Hodge theory of $p$-adic varieties: a survey

$p$-adic Hodge theory is one of the most powerful tools in modern arithmetic geometry. In this survey, we will review $p$-adic Hodge theory of algebraic varieties, present current developments in $p$-adic Hodge theory of analytic varieties, and discuss some of its applications to problems in number theory. This is an extended version of a talk at the Jubilee Congress for the 100th anniversary of the Polish Mathematical Society, Kraków, 2019.