The Hochschild cohomology of an algebra is itself both an associative algebra and a graded Lie algebra. We will begin by introducing Hochschild cohomology and these structures. Then we will discuss some recent results leading to better understanding of the Lie structure. We will focus on Volkov's homotopy lifting method for defining brackets by way of an arbitrary resolution for algebras over a field, and methods developed with Negron for computing brackets on resolutions that are differential graded coalgebras. We will introduce $A_{\infty}$-coalgebras as a larger framework for some of these results, and make a connection to Stasheff's realization of the Lie bracket as a graded commutator of coderivations on a tensor coalgebra.