The purpose of these notes is to survey results concerning the formation of codimension one interfaces for multi-well gradient-driven problems. The theory is well advanced in the scalar case, where the equation is often referred to as the Allen-Cahn equation, but is still widely open in the vectorial case. After a discussion on so called slow motion in dimension one (where interfaces reduce to points and the limiting dynamics to ordinary differential equations), we turn to the stationary case in higher dimensions, where the interfaces are expected to be weak minimal surface. The proofs for the scalar case rely for a large part on a monotonicity formula for the energy density, which is itself related to the vanishing of the so-called discrepancy function. The vectorial case in contrast is quite open. This lack of results and insight is to a large extend related to the absence of known appropriate monotonicity formula. In the last parts, we focus on the elliptic case in two dimensions, and introduce methods, relying on the analysis of the partial differential equation, which allow to circumvent the lack of monotonicity formula for the energy density.