We study the metric dimension (strong and weak) of infinite graphs. In particular, our main interest is characterizing infinite graphs with finite dimension. Our main results: (1) graphs with more than one end have infinite strong dimension; (2) for graphs with a finite number of cycles, the weak dimension is finite if and only if the graph has finitely many vertices of degree three, and the strong dimension is finite if and only if the graph has one end and finitely many vertices of degree three.