Period domains D can be described as certain analytic open sets of flag varieties; due to the presence of monodromy, however, the period map of a family of algebraic varieties lands in a quotient D/\Gamma by an arithmetic group.  In the very special case when D/\Gamma is itself algebraic, understanding the interaction between algebraic structures on the source and target of the uniformization D\rightarrow D/\Gamma is a crucial component of the modern approach to the André-Oort conjecture.  We prove a version of the Ax-Schanuel conjecture for general period maps X\rightarrow D/\Gamma which says that atypical algebraic relations between X and D are governed by Hodge loci.  We will also discuss some geometric and arithmetic applications.  This is joint work with J. Tsimerman.