This thesis has two main parts. The rst part, consisting of two papers, concerns the algebraic structure on Hochschild complexes of commutative Hopf algebras and their weaker cousins, such as commutative quasi-Hopf algebras and commutative Hopsh algebras. For any of the above, we equip the Hochschild complex with a natural Hopf algebra structure up to coherent homotopy. In the rst paper, we study the interplay between the Hochschild complex and the Dold-Kan equivalence between connective chain complexes and simplicial modules over a commutative ring. As an application, we obtain a strictication of the coherent commutative Hopf algebra structure on the Hochschild complex of a commutative Hopf algebra. In the second paper we study the functoriality of the Hochschild complex with respect to bimodules. This allows us to upgrade the Hochschild complex to a symmetric monoidal functor of quasi-categories from a certain nerve of the (2,1)-category of bimodules between algebras to the quasi-category of chain complexes. Using the fact that certain families of Hopf-like algebras are special cases of Hopsh algebras, we obtain as an application that the Hochschild complexes of such algebras have a natural Hopf algebra structure up to coherent homotopy. The second part of the thesis is a work in progress, generalizing the work of Wahl and Westerland on operations on Hochschild complexes to construct operations on topological Hochschild homology. Our main theorem, conditioned on a technical quasi-category-theoretical conjecture, is the construction of an action of moduli spaces of Riemann surfaces on the topological Hochschild homology of A∞-Frobenius algebras.