  We develop a theory of the direct interband and indirect intraband
photogalvanic effects in Weyl semimetals belonging to the gyrotropic classes
with improper symmetry operations. At zero magnetic field, an excitation of
such a material with circularly polarized light leads to a photocurrent whose
direction depends on the light helicity. We show that in the semimetals of the
C$_{2v}$ symmetry, an allowance for the tilt term in the effective Hamiltonian
is enough to prevent cancellation of the photocurrent contributions from the
Weyl cones of opposite chiralities. In the case of the C$_{4v}$ symmetry, in
addition to the tilt it is necessary to include terms of the second- or
third-order in the electron quasi-momentum. For indirect intraband transitions,
the helicity-dependent photocurrent generated within each Weyl node takes on a
universal value determined by the fundamental constants, the light frequency
and electric field. We have complementarily investigated the magneto-gyrotropic
photogalvanic effect, i.e. an appearance of a photocurrent under unpolarized
excitation in a magnetic field. In quantized magnetic fields, the photocurrent
is caused by optical transitions between the one-dimensional magnetic subbands.
A value of the photocurrent is particularly high if one of the photocarriers is
excited to the chiral subband with the energy below the cyclotron energy.
