  We construct SU($N$) super Yang-Mills theories with extended supersymmetry on
hypercubic lattices of various dimensions keeping one or two supercharges
exactly.
  It is based on topological field theory formulation for the super Yang-Mills
theories. Gauge fields are represented by compact unitary link variables, and
the exact supercharges on the lattice are nilpotent up to gauge
transformations. In particular, the lattice models are free from the vacuum
degeneracy problem, which was encountered in earlier approaches. Thus, we do
not need to introduce any supersymmetry breaking terms, and the exact
supersymmetry is preserved wholly in the process of taking the continuum limit.
  Among the models, we show that the desired continuum theories are obtained
without any fine tuning of parameters for the cases ${\cal N}=2, 4, 8$ in
two-dimensions. Also, the cases ${\cal N}=4, 8$ in three-dimensions are
investigated, and a problem arising in four-dimensional models is discussed.
