We have studied the critical properties of the three-dimensional random anisotropy Heisenberg model by means of numerical simulations using the Parallel Tempering method. We have simulated the model with two different disorder distributions, cubic and isotropic ones, with two different anisotropy strengths for each disorder class. We have obtained clear signatures of a second order phase transition (paramagnetic-ferromagnetic) for the cubic and the isotropic disorder distributions and we have characterized the critical exponents and cumulants. In the case of isotropic disorder distribution we have found a strong dependence of the thermal exponent on the disorder strength. Finally, we have found evidences of universality for the case of the anisotropic disorder distribution by finding critical exponents and universal dimensionless ratios independent of the strength of the disorder.