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 Tsygan-Tamarkin), in particular 
the BV structure in the Calabi-Yau case

Relation to the cotangent complex/ Andre-Quillen 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 K-theory

topological Hochschild and cyclic homology, the cyclotomic trace, $K^S=THH$

HH for E_n algebras and the Deligne-Kontsevich 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...)