  We present an axiomatic approach to finite- and infinite-dimensional
differential calculus over arbitrary infinite fields (and, more generally,
suitable rings). The corresponding basic theory of manifolds and Lie groups is
developed. Special attention is paid to the case of mappings between
topological vector spaces over non-discrete topological fields, in particular
ultrametric fields or the fields of real and complex numbers. In the latter
case, a theory of differentiable mappings between general, not necessarily
locally convex spaces is obtained, which in the locally convex case is
equivalent to Keller's C^k_c-theory.
