Blackstaff's Tome of Wizardry
Blackstaff's Tome of Wizardry is a splatbook for Dungeons & Dragons 5th Edition that seeks to expand the Wizard class with a whopping 31 new subclasses to represent different arcane traditions. It also includes new character personalizing tables to reflect your wizard PC's belief in the ultimate source of magic and how they first became a wizard, a plethora of new spells to supplement each subclass, and a 5e conversion of the Guardinal race of Celestials, which has been missing in action since the end of 3rd edition.
School of Alchemical Artillery
This is a unique blend of an alchemist and an evoker, focusing on utilising arcane magic combined with alchemical formula to a maximise their abilities to unleash a strong offense from a long range.
Arcane Experimenter
One of several subclasses in this splat that technically harken back to the "generalist" wizard of old, the Arcane Experimenter is a "practical" wizard who, as their name suggests, is constantly experimenting with new ways to cast or manipulate magic, and goes into battle to test out their latest formula. The result is a wizard subclass with a set of almost fighter-like focus on a unique set of spellcasting tricks.
School of Astromancy
This is a practitioner of magic drawing from the powers of the stars and planets... in Warhammer Fantasy terms, this is a wizard who practices the Lore of the Heavens.
School of Battlemagi
You might be thinking that this is a revamped warmage, right? Nope! Instead, it's a revival of the swordmage, attempting to be more faithful to its unique 4e depiction as opposed to the older, Forgotten Realms-centric Bladesinger.
School of Binding
As you can probably tell from the name of this one, it's an attempt to bring back the spirit of the binder. These specialist conjurers have unique subclass abilities that focus on their ability to summon forth a planar entity - an angel, guardinal, demon or devil - and physically bind it into their flesh in the form an enchanted tattoo.
School of Dimensionalism=
This the long-forgotten Dimensionalist brought back into 5th edition! These are specialist conjurers who focus on manipulation dimensional barriers and portals, rather than summoning monsters.
Elemental Traditions
Obviously, these are the elementalism subclasses; you couldn't expect a splatbook like this to lack something as obvious yet untapped as that! This book handles each form of elementalism as its own distinct subclass.
School of Aeromancy
This is the Air Elementalist, obviously.
School of Floralmancy
Naturally often confused with druids, floramancy wizards hearken to Chinese Elementalism, being the Wood Elementalist subclass.
School of Geomancy
As you can probably tell, this is the Earth Elementalist.
School of Hydromancy
No prizes for guessing that this is the Water Elementalist, unfortunately.
School of Pyromancy
Yes, this is the Fire Elementalist. Pat yourself on the back.
School of Emomancy
No, this is not connected to bad 90s bardic traditions. Emomantics, as they call themselves, are a variant form of enchanter who specialize in using their magic to manipulate and control emotions in themselves and others. This one comes with an "authorial sidebar" noting that, despite having a better reputation than necromancers, enchanters are actually pretty damn creepy if not evil when you think about it.
School of Ferromancy
Despite not being placed under the Elemental Traditions section for whatever reason, this is effectively the Metal Elementalist, with magic focusing on manipulating metal and magnetic fields.
School of Fleuremancy
Arguably, this school is best summarized as the Light Elementalist; it's a wizard who focuses on mastering the powers and applications of radiant energy.
School of Golemancy
This is a wizard focused on the arts of crafting golems; it has no direct precedents amongst the "core eight" schools of magic, but is arguably closest in thematic elements to the necromancer.
The Green Star Adept
A blast from the past, the Green Star Adept first appeared in the Complete Arcane as a gish-themed prestige class that slowly transformed the character into a living golem of luminescent green "star-metal". It was widely considered terrible. This subclass is basically a 5eification of that original prestige class, and preserves the original concept: you're a wizard who seeks to fundamentally empower yourself with magical energy by ingesting alchemical formulas based around the viridian-hued star-metal left behind by the Green Star of Alhazarde.
Guild Wizardry
As the name suggests, this is a more "generalist" wizard archetype, specifically focused on representing a wizard who belongs to an established wizard guild that has a sufficiently formal hierarchy to allow for beneficial studies and social connections.
School of Hedge Magi
In comparison to the Guild Wizard, the Hedge Mage is a self-taught practitioner of the arcane arts who, in many ways, exemplifies the adventurer-wizard, with a focus on self-reliance and versatility.
School of Incantation
This is another blast from the past, in the form of the Metamage - or, as it's known in the Forgotten Realms, the Incanter. This is a wizard focused on manipulating the raw magical potential in themselves and others, making them masters in augmenting and suppressing spells and a real nightmare for other wizards to face.
School of Jaunting
Self-admittedly described in the book as a strange blending of the "surface traditions" of the Dimensionalist, Chronomancy and the "Translocation" aspect of Conjuration, Jaunters are wizards specialized in manipulating space and time to unparalleled extents.
School of Kinetics
Overlapping to a degree with the psion, the Kineticist, as you can probably guess from the name, is focused on using magic to manipulate space and to move objects. Basically, this school specializes in telekinesis, teleportation and other forms of advanced movement alteration.
School of Lunar Magi
Arguably a specialized splinter school of the Astromancer, the Lunar Mage is, as the name implies, a wizard who focuses on drawing magical energies directly from the moon.
School of Mentalism
You might think that this is a psion using a wizard's chassis and... well, you're not exactly wrong, but in fact these sub-specialist enchanters are actually a real old-school wizard subclass from back in the days of AD&D. This version openly admits that it straddles the line between the wizard and the psion.
School of Nethermancy
This is the iconic practitioner of Shadow Magic, brought forth into 5th edition at long last.
School of Puppetry
This is a variant Golemancer (see above) who focuses on animating small but more versatile puppets to serve their will, rather than the more traditional big hulking golem.
School of Reanimation
The specialist necromancers are particularly focused on the idea of how necromancy can be used to restore life to the fallen, and thusly specialized in necromantic healing, lifeforce manipulation, and the creation of revenants, homunculi and flesh golems.
School of Runecraft
Rune magic has appeared in D&D more than once - just check out the Runemaster, Runecaster and the Runepriest. The Runecrafter differs from all of those in that it focuses on inscribing enchanted runes into their flesh as magical tattoos, scars or brands, giving them a number of unique innate magical abilities depending on which runes they choose to adorn themselves with.
School of Scrymancy
These are the self-described spies of wizardry, and more or less what you get when you blend a rogue and a diviner.
School of Soulbinding
Arguably a specialist form of necromancer, the Soulbinder uses ritual magic to bind a portion of their own life force into chosen items, creating powerful arcane focuses that can also be used as separate wellsprings of vital energies - just the thing to make the Soulbinder a little bit harder to kill than the average wizard!
Symbarch Teachings
The only tradition in this splatbook that's super-connected to the Forgotten Realms (albeit the Metamage debuted in an FR splatbook), this represents a wizard who practices the unique magical style/tradition passed down by the Simbul, the Witch-Queen of Aglarond and one of the Seven Sisters. The book also contains a page-long summary of the Simbul's history in the Forgotten Realms.
School of Voidmancy
Voidmancers are conjurers who focus on studying the dark void on the edge of reality from whence aberrations hail - essentially, the "safer" margins of the Far Realm - and who learn to gain heightened mental power from doing so without going nuts in the process.