We survey some recent results on the gradient flow of an anisotropic surface en-ergy, the integrand of which is one-homogeneous in the normal vector. We discuss the reasonsfor assuming convexity of the anisotropy, and we review some known results in the smooth,mixed and crystalline case. In particular, we recall the notion of calibrability and the relatedfacet-breaking phenomenon. Minimal barriers as weak solutions to the gradient flow in case ofnonsmooth anisotropies are proposed. Furthermore, we discuss some relations between cylin-drical anisotropies, the prescribed curvature problem and the capillarity problem. We concludethe paper by examining some higher order geometric functionals. In particular we discuss theanisotropic Willmore functional and compute its first variation in the smooth case.
