This talk is meant to give an introduction and overview on the theme of conformal field theories and their geometric aspects. We start with a brief review of approaches to conformal field theory, both axiomatically and motivated from geometry by means of the chiral de Rham complex. Recalling toroidal conformal field theories, we will find examples of conformal field theories where the geometric content is particularly clear. We explain how these theories can be used to obtain explicit constructions of conformal field theories along with concrete geometric interpretations on more complicated geometries, like K3 surfaces and certain Calabi-Yau threefolds.