Arcane Recovery
PHB p112
You have learned to regain some of your magical energy by studying your spellbook. Once per day when you finish a short rest, you can choose expended spell slots to recover. The spell slots can have a combined level that is equal to or less than half your wizard level (rounded up), and none of the slots can be 6th level or higher.

For example, if you're a 4th-level wizard, you can recover up to two levels worth of spell slots.

You can recover either a 2nd-level spell slot or two 1st-level spell slots.


Spellcasting
PHB p112
As a student of arcane magic, you have a spellbook containing spells that show the first glimmerings of your true power. See chapter 10 for the general rules of spellcasting and chapter 11 for the wizard spell list.

Cantrips
At 1st level, you know three cantrips of your choice from the wizard spell list. You learn additional wizard cantrips of your choice at higher levels, as shown in the Cantrips Known column of the Wizard table.

Spellbook
At 1st level, you have a spellbook containing six 1st-level wizard spells of your choice. Your spellbook is the repository of the wizard spells you know, except your cantrips, which are fixed in your mind.

Preparing and Casting Spells
The Wizard table shows how many spell slots you have to cast your spells of 1st level and higher. To cast one of these spells, you must expend a slot of the spell's level or higher. You regain all expended spell slots when you finish a long rest.

You prepare the list of wizard spells that are available for you to cast. To do so, choose a number of wizard spells from your spellbook equal to your Intelligence modifier + your wizard level (minimum of one spell). The spells must be of a level for which you have spell slots.

For example, if you're a 3rd-level wizard, you have four 1st-level and two 2nd-level spell slots. With an Intelligence of 16, your list of prepared spells can include six spells of 1st or 2nd level, in any combination, chosen from your spellbook. If you prepare the 1st-level spell magic missile, you can cast it using a 1st-level or a 2nd-level slot. Casting the spell doesn't remove it from your list of prepared spells.

You can change your list of prepared spells when you finish a long rest. Preparing a new list of wizard spells requires time spent studying your spellbook and memorizing the incantations and gestures you must make to cast the spell: at least 1 minute per spell level for each spell on your list.

Spellcasting Ability
Intelligence is your spellcasting ability for your wizard spells, since you learn your wizard spells through dedicated study and memorization. You use your Intelligence whenever a spell refers to your spellcasting ability. In addition, you use your Intelligence modifier when setting the saving throw DC for a wizard spell you cast and when making an attack roll with one.

Spell save DC = 8 + your proficiency bonus + your Intelligence modifier
Spell attack modifier = your proficiency bonus + your Intelligence modifier
Ritual Casting
You can cast a wizard spell as a ritual if that spell has the ritual tag and you have the spell in your spellbook. You don't need to have the spell prepared.

Spellcasting Focus
You can use an arcane focus as a spellcasting focus for your wizard spells.

Learning Spells of 1st Level and Higher
Each time you gain a wizard level, you can add two wizard spells of your choice to your spellbook. Each of these spells must be of a level for which you have spell slots, as shown on the Wizard table. On your adventures, you might find other spells that you can add to your spellbook (see "Your Spellbook").

Your Spellbook
The spells that you add to your spellbook as you gain levels reflect the arcane research you conduct on your own, as well as intellectual breakthroughs you have had about the nature of the multiverse. You might find other spells during your adventures. You could discover a spell recorded on a scroll in an evil wizard's chest, for example, or in a dusty tome in an ancient library.

A spellbook doesn't contain cantrips.

Copying a Spell into the Book. When you find a wizard spell of 1st level or higher, you can add it to your spellbook if it is of a spell level you can prepare and if you can spare the time to decipher and copy it.
Copying a spell into your spellbook involves reproducing the basic form of the spell, then deciphering the unique system of notation used by the wizard who wrote it. You must practice the spell until you understand the sounds or gestures required, then transcribe it into your spellbook using your own notation.

For each level of the spell, the process takes 2 hours and costs 50 gp. The cost represents material components you expend as you experiment with the spell to master it, as well as the fine inks you need to record it. Once you have spent this time and money, you can prepare the spell just like your other spells.

Copying from a Spell Scroll. A wizard spell on a spell scroll can be copied just as spells in spellbooks can be copied. When you copy a spell from a spell scroll, you must succeed on an Intelligence (Arcana) check with a DC equal to 10 + the spell's level. If the check succeeds, the spell is successfully copied. Whether the check succeeds or fails, the spell scroll is destroyed.
Replacing the Book. You can copy a spell from your own spellbook into another book—for example, if you want to make a backup copy of your spellbook. This is just like copying a new spell into your spellbook, but faster and easier, since you understand your own notation and already know how to cast the spell. You need spend only 1 hour and 10 gp for each level of the copied spell.
If you lose your spellbook, you can use the same procedure to transcribe the spells that you have prepared into a new spellbook. Filling out the remainder of your spellbook requires you to find new spells to do so, as normal. For this reason, many wizards keep backup spellbooks in a safe place.

The Book's Appearance. Your spellbook is a unique compilation of spells, with its own decorative flourishes and margin notes. It might be a plain, functional leather volume that you received as a gift from your master, a finely bound gilt-edged tome you found in an ancient library, or even a loose collection of notes scrounged together after you lost your previous spellbook in a mishap.

Arcane Tradition
PHB p112
When you reach 2nd level, you choose an arcane tradition from the list of available traditions, shaping your practice of magic. Your choice grants you features at 2nd level and again at 6th, 10th, and 14th level.

Bladesinging
TCE p76
Bladesingers are elves who bravely defend their people and lands. They are elf wizards who master a school of sword fighting grounded in a tradition of arcane magic. In combat, a bladesinger uses a series of intricate, elegant maneuvers that fend off harm and allow the bladesinger to channel magic into devastating attacks and a cunning defense.

Training in War and Song (Bladesinging)
When you adopt this tradition at 2nd level, you gain proficiency with light armor, and you gain proficiency with one type of one-handed melee weapon of your choice.

You also gain proficiency in the Performance skill if you don't already have it.

Bladesong
Starting at 2nd level, you can invoke a secret elven magic called the Bladesong, provided you aren't wearing medium or heavy armor or using a shield. It graces you with supernatural speed, agility, and focus.

You can use a bonus action to start the Bladesong, which lasts for 1 minute. It ends early if you are incapacitated, if you don medium or heavy armor or a shield, or if you use two hands to make an attack with a weapon. You can also dismiss Bladesong at any time you choose (no action required).

While your bladesong is active, you gain the following benefits:

You gain a bonus to your AC equal to your Intelligence modifier (minimum of +1).
Your walking speed increases by 10 feet.
You have advantage on Dexterity (Acrobatics) checks.
You gain a bonus to any Constitution saving throws you make to maintain concentration on a spell. The bonus equals your Intelligence modifier (minimum of +1).
You can use this feature a number of times equal to your proficiency bonus, and you regain all expended uses of it when you finish a long rest.

Bladesinger Styles
From its inception as a martial and magical art, Bladesinging has been tied to the sword, more specifically the longsword. Yet many generations of study gave rise to various styles of Bladesinging based on the melee weapon employed. The techniques of these styles are passed from master to students in small schools, some of which have a building dedicated to instruction. Even the newest styles are hundreds of years old, but are still taught by their original creators due to the long lives of elves. Most schools of Bladesinging are in Evermeet or Evereska. One was started in Myth Drannor, but the city's destruction has scattered those students who survived.

Styles of Bladesinging are broadly categorized based on the type of weapon employed, and each is associated with a category of animal. Within that style are specializations named after specific animal types, based on the types of spells employed, the techniques of the master, and the particular weapon used. Bladesingers who apprentice to a master typically get a tattoo of their chosen style's animal. Some bladesingers learn multiple styles and bear many tattoos, wearing a warning on their skin of their deadly skills.

Cat. Styles that employ a sword belong to this family. The lion style, the eldest, trains practitioners in the use of the longsword and doesn't favor any particular type of spells. Leopard style focuses on the shortsword and spells of illusion and stealth. Red tiger, a style just three centuries old, has its bladesingers using the scimitar in a whirling dance of defense from which they launch into sudden leaps and attacks.
Bird. Styles that focus on the use of a hafted weapon, such as an axe or hammer, have been grouped together as bird styles, yet they vary wildly. All relatively new styles, they use weapons not typically favored by elves. Eagle-style bladesingers use small handaxes, and many maneuvers in the style focus on fluid ways to throw the weapon and draw a new one. Raven style uses a war pick, and spells associated with it grant the bladesinger more agility in combat.
Snake. Practitioners of these styles use a flail, chain, or whip. Viper style uses a whip, despite its inelegance as a weapon, and has almost as long a history as the lion style. Its masters punctuate their bladesong with a stunningly rapid rhythm of whip cracks, which can keep many foes at bay and allow the bladesinger space to cast the cruel spells of poison and disease favored by the style.
Subclass source: TCE, page 76. Also found in SCAG, page 141.

Ability Score Improvement
PHB p112
When you reach 4th level, you can increase one ability score of your choice by 2, or you can increase two ability scores of your choice by 1. As normal, you can't increase an ability score above 20 using this feature.

If your DM allows the use of feats, you may instead take a feat.

Unending Breath. You can hold your breath indefinitely while you're not incapacitated.

Mingle with the Wind. You can cast the levitate spell once with this trait, requiring no material components, and you regain the ability to cast it this way when you finish a long rest. Constitution is your spellcasting ability for this spell.

All Eyes on You
Your accent, mannerisms, figures of speech, and perhaps even your appearance all mark you as foreign. Curious glances are directed your way wherever you go, which can be a nuisance, but you also gain the friendly interest of scholars and others intrigued by far-off lands, to say nothing of everyday folk who are eager to hear stories of your homeland.

You can parley this attention into access to people and places you might not otherwise have, for you and your traveling companions. Noble lords, scholars, and merchant princes, to name a few, might be interested in hearing about your distant homeland and people.

Spellbook, scholar's pack, Arcane Focus, Dagger, Scimitar, traveler's clothes, Lute, poorly wrought maps from your homeland that depict where you are in Faerûn, a small piece of jewelry worth 10 gp in the style of your homeland's craftsmanship, and a pouch containing 505gp, infernal obsidian crystals that absorbs light, Stone of luck: +1 to ability checks and saving throws while being carried on your person. Requires attunment; Potion of cure light wounds.

I begin or end my day with small traditional rituals that are unfamiliar to those around me.

Adventure. I'm far from home, and everything is strange and wonderful! (Chaotic)

My freedom is my most precious possession. I'll never let anyone take it from me again.

I pretend not to understand the local language in order to avoid interactions I would rather not have.