By making use of the Kubo formula, we calculate the conductivity of Dirac and Weyl semimetals in a magnetic field. We find that the longitudinal (along the direction of the magnetic field) magnetoresistivity is negative at sufficiently large magnetic fields for both Dirac and Weyl semimetals. The physical reason of this phenomenon is intimately connected with the dimensional spatial reduction 3→1 in the dynamics of the lowest Landau level. The off-diagonal component of the transverse (with respect to the direction of the magnetic field) conductivity in Weyl semimetals contains an anomalous contribution directly proportional to the momentum-space separation between the Weyl nodes. This contribution comes exclusively from the lowest Landau level and, as expected, is independent of the temperature, chemical potential, and magnetic field. The other part of the off-diagonal conductivity is the same as in Dirac semimetals and is connected with a nonzero density of charge carriers. The signatures for experimental distinguishing Weyl semimetals from Dirac ones through the measurements of conductivity are discussed.