Abstract: The study of period polynomials for classical modular forms has emerged due to their role in Eichler cohomology. In particular, the Eichler-Shimura isomorphism gives a correspondence between cusp eigenforms and their period polynomials. The coefficients of period polynomials also encode critical L-values for the associated modular form and thus contain rich arithmetic information. In this talk, we will examine period polynomials from both angles, including their cohomological interpretation as well as some of their analytic properties. Finally, I will describe joint work with Larry Rolen and Ian Wagner, where we introduce period polynomials for Hilbert modular forms of full level and prove that their zeros lie on the unit circle.