Algebraization theorems originating from o-minimality have found striking applications in recent years to Hodge theory and Diophantine geometry. The utility of o-minimality originates from the 'tame' topological properties that sets definable in such structures satisfy. O-minimal geometry thus provides a way to interpolate between the algebraic and analytic worlds. One such algebraization theorem that has been particularly useful is the definable Chow theorem of Peterzil and Starchenko which states that a closed analytic subset of a complex algebraic variety that is simultaneously definable in an o-minimal structure is an algebraic subset. In this talk, I shall discuss a non-archimedean version of this result and some recent applications of these algebraization theorems.