This thesis details recent results on the periodic homogenization of interface motions in the parabolic scaling regime. First, we treat phase transitions in periodic media, developing a variational approach that unifies the point of view of Barles and Souganidis with Aubrey-Mather theory. We show that (possibly non-smooth) pulsating standing waves can be obtained as minimizers of a Percival-type Lagrangian in the spirit of the latter theory. Pulsating standing waves are shown to generate the recurrent plane-like minimizers of the energy and to determine the macroscopic surface tension. In the case of laminar media, these functions are used to demonstrate a number of pathologies, such as the non-differentiability of the surface tension. Additionally, we prove two homogenization results in the sharp-interface limit, one pertaining to laminar media and a restricted class of initial data, and the other, to the Allen-Cahn equation with a periodic dissipation term. Finally, we study a class of non-variational curvature flows with periodic coefficients. By analyzing a degenerate elliptic equation on the torus, we identify the macroscopic behavior of the moving interface at points where its normal vector is irrational. To analyze the motion near rational normals, we adapt an idea of Feldman and Kim from the study of oscillating boundary value problems. This leads to the conclusion that in dimensions three and higher, the macroscopic interface velocity is a discontinuous function of the normal direction. We prove that the associated interface motion is well-posed in spite of the discontinuities and conclude that homogenization occurs.