In classical Hodge theory, variations of Hodge structure and their associated period mappings play a crucial role. In the $p$-adic world, it turns out there are *two* natural kinds of period maps associated with variations of $p$-adic Hodge structure: the ``Grothendieck-Messing" period maps, which roughly come from comparing crystalline and de Rham cohomology, and the ``Hodge-Tate" period maps, which come from comparing de Rham and $p$-adic etale cohomology. I'll discuss these period maps, their applications, and some new results on their construction and geometry. This is partially joint work with Jared Weinstein.