  We present results of a theoretical study of photocurrents in the Weyl
semimetals belonging to the gyrotropic symmetry classes. We show that, in
weakly gyrotropic symmetry classes C$_{nv}$ ($n = 3,4,6$), the circular
photocurrent transverse to the incidence direction appears only with account,
in the electron effective Hamiltonian, for both linear and quadratic or cubic
in quasi-momentum spin-dependent terms as well as a spin-independent term
resulting in the tilt of the cone dispersion. A polarization-independent
magneto-induced photocurrent is predicted which is also allowed in gyrotropic
systems only. For crystals of the C$_{2v}$ symmetry, we consider a microscopic
mechanism of the photocurrent in a quantized magnetic field which is generated
under direct optical transitions between the ground and the first excited
magnetic subbands. It is shown that this current becomes nonzero with allowance
for anisotropic tilt of the dispersion cones.
