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