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.