Hochschild homology of a ring has a topological analogue for ring spectra, topological Hochschild homology (THH), which plays an essential role in the trace method approach to algebraic K-theory. Using norms in equivariant stable homotopy theory, one can define a twisted version of THH, which takes as input an equivariant ring spectrum. In this talk I will discuss categorical approaches to studying twisted THH, as well as computational tools. In particular, I will discuss the twisted THH of Thom spectra and the Real bordism spectrum. This talk includes joint work with Angeltveit, Blumberg, Hill, Lawson, and Mandell, as well as joint work with Adamyk, Hess, Klang, and Kong.