  We discuss criteria for the nonexistence, existence and computation of
invariant algebraic surfaces for three-dimensional complex polynomial vector
fields, thus transferring a classical problem of Poincar\'e from dimension two
to dimension three. Such surfaces are zero sets of certain polynomials which we
call semi-invariants of the vector fields. The main part of the work deals with
finding degree bounds for irreducible semi-invariants of a given polynomial
vector field that satisfies certain properties for its stationary points at
infinity. As a related topic, we investigate existence criteria and properties
for algebraic Jacobi multipliers. Some results are stated and proved for
polynomial vector fields in arbitrary dimension and their invariant
hypersurfaces. In dimension three we obtain detailed results on possible degree
bounds. Moreover by an explicit construction we show for quadratic vector
fields that the conditions involving the stationary points at infinity are
generic but they do not a priori preclude the existence of invariant algebraic
surfaces. In an appendix we prove a result on invariant lines of homogeneous
polynomial vector fields.
