  Until now, the only known maximal surfaces in Minkowski 3-space of finite
topology with compact singular set and without branch points were either genus
zero or genus one, or came from a correspondence with minimal surfaces in
Euclidean 3-space given by the third and fourth authors in a previous paper. In
this paper, we discuss singularities and several global properties of maximal
surfaces, and give explicit examples of such surfaces of arbitrary genus. When
the genus is one, our examples are embedded outside a compact set. Moreover, we
deform such examples to CMC-1 faces (mean curvature one surfaces with
admissible singularities in de Sitter 3-space) and obtain "cousins" of those
maximal surfaces.
  Cone-like singular points on maximal surfaces are very important, although
they are not stable under perturbations of maximal surfaces. It is interesting
to ask if cone-like singular points can appear on a maximal surface having
other kinds of singularities. Until now, no such examples were known. We also
construct a family of complete maximal surfaces with two complete ends and with
both cone-like singular points and cuspidal edges.
