We survey some recent results on the gradient flow of an anisotropic surface energy, the integrand of which is one-homogeneous in the normal vector. We discuss the reasons for 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 related facet-breaking phenomenon. Minimal barriers as weak solutions to the gradient flow in case of nonsmooth anisotropies are proposed. Furthermore, we discuss some relations between cylindrical anisotropies, the prescribed curvature problem and the capillarity problem. We conclude the paper by examining some higher order geometric functionals. In particular we discuss the anisotropic Willmore functional and compute its first variation in the smooth case.