  This paper aims at building a unified framework to deal with a wide class of
local and nonlocal translation-invariant geometric flows. First, we introduce a
class of generalized curvatures, and prove the existence and uniqueness for the
level set formulation of the corresponding geometric flows.
  We then introduce a class of generalized perimeters, whose first variation is
an admissible generalized curvature. Within this class, we implement a
minimizing movements scheme and we prove that it approximates the viscosity
solution of the corresponding level set PDE.
  We also describe several examples and applications. Besides recovering and
presenting in a unified way existence, uniqueness, and approximation results
for several geometric motions already studied and scattered in the literature,
the theory developed in this paper allows us to establish also new results.
