  For a classical group over a non-archimedean local field of odd residual
characteristic p, we construct all cuspidal representations over an arbitrary
algebraically closed field of characteristic different from p, as
representations induced from a cuspidal type. We also give a fundamental step
towards the classification of cuspidal representations, identifying when
certain cuspidal types induce to equivalent representations; this result is new
even in the case of complex representations. Finally, we prove that the
representations induced from more general types are quasi-projective, a crucial
tool for extending the results here to arbitrary irreducible representations.
