We present a rigorous formalism to solve the problem of diffraction of light at a periodically corrugated boundary between an isotropic medium (dielectric or metal with losses) and a biaxial crystal. The method applies to gratings illuminated either from the isotropic or from the biaxial side by waves with wave vectors inclined at an arbitrary angle with respect to the grooves and for arbitrary orientations of the crystal optic axes. Using a nonorthogonal curvilinear coordinate transformation that simplifies the boundary conditions at the grating interface and writing Maxwell's equations for the covariant components of the fields in the transformed frame, the problem can be reduced to the numerical solution of a system of first order differential equations with constant coefficients. The application of the method is illustrated in two cases: (i) diffraction of s- and p-polarized waves at a sinusoidal boundary between a transparent dielectric and a biaxial crystal and (ii) excitation of surface plasmons along the corrugated interface between a metal and a biaxial crystal.