  We discuss in general the construction of gauge-invariant non-local meson
operators on the lattice. We use such operators to study the $P$- and $D$-wave
mesons as well as hybrid mesons in quenched QCD, with quark masses near the
strange quark mass. The resulting spectra are compared with experiment for the
orbital excitations. For the states produced by gluonic excitations (hybrid
mesons) we find evidence of mixing for non-exotic quantum numbers. We give
predictions for masses of the spin-exotic hybrid mesons with $J^{PC}=1^{-+},\
0^{+-}$, and $2^{+-}$.
