Classical Clark measures are singular measures on the unit circle defined via inner functions that are closely tied to important topics in operator theory and complex analysis (for example, model spaces, compressed shifts, and composition operators). In this talk, we’ll consider an analogous definition for Clark measures associated with two-variable inner functions. For certain classes of such functions, we’ll give exact formulas for these Clark measures, characterize when associated Clark embeddings are unitary, and obtain nice unitary perturbations of pairs of compressed shift operators. This is joint work with John Anderson, Palak Arora, Linus Bergqvist, Joseph Cima, Conni Liaw, and Alan Sola.