  We describe a natural strategy to enumerate compact hyperbolic 3-manifolds
with geodesic boundary in increasing order of complexity. We show that the same
strategy can be employed to analyze simultaneously compact manifolds and
finite-volume manifolds having toric cusps. In opposition to this we show that,
if one allows annular cusps, the number of manifolds grows very rapidly, and
that our strategy cannot be employed to obtain a complete list. We also
carefully describe how to compute the volume of our manifolds, discussing
formulae for the volume of a tetrahedron with generic dihedral angles in
hyperbolic space.
