There is currently no treatise treating Hochschild (co)homology systematically. There is a chapter in Weibel's book, there's parts of Loday's and a few others...
What should be covered by such a mythical treatise?
There is currently no treatise treating Hochschild (co)homology systematically. There is a chapter in Weibel's book, there's parts of Loday's and a few others...



My wishlist: Hochschild (co)homology of (curved) Ainfinity algebras/categories Relation to deformation theory HochschildKostantRosenberg Hochschildcyclic spectral sequence and relation to Hodgede Rham spectral sequence Deligne conjecture Relation to Drinfeld center 2D TQFTs (Costello/Kontsevich/HopkinsLurie) 


In addition to Kevin's excellent list: Formality The relation of Hochschild and cyclic homology with loop spaces (eg Jones' theorem) and the circle action on Hochschild homology operadic structure of $(HH^\ast,HH_*)$ (ie "calculus" a la TsyganTamarkin), in particular the BV structure in the CalabiYau case Relation to the cotangent complex/ AndreQuillen homology in the commutative case The role of Hochschild homology as recipient of characters (eg Chern characters and characters of representations)  more generally the relation with algebraic Ktheory topological Hochschild and cyclic homology, the cyclotomic trace, $K^S=THH$ HH for E_n algebras and the DeligneKontsevich conjecture Lie theoretic perspective ($HH^\ast$ as universal enveloping algebra of the Atiyah bracket on the shifted tangent complex, HKR theorem as PBW, $HH^*$ as the Lie algebra of autoequivalences of the derived category...) 


A personal list.



I am working on producing an account from a modern perspective at Hochschild cohomology on the $n$Lab. Some of the wishlist items expressed here are already being covered to some extent. But clearly more needs to be done. 


I really want to know more about Hochschild cohomology of commutative algebras, and its relation to the representations of $S_n$ and to free Lie algebras (beyond "there are these strange idempotents in $\mathbb Q\left[S_n\right]$ which happen to occur in both fields). 

