  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. For the case of the anisotropic
disorder, we have found evidences of universality by finding critical exponents
and universal dimensionless ratios independent of the strength of the disorder.
In the case of isotropic disorder distribution the situation is very involved:
we have found two phase transitions in the magnetization channel which are
merging for larger lattices remaining a zero magnetization low temperature
phase. Studying this region using a spin glass order parameter we have found
evidences for a spin glass phase transition. We have estimated effective
critical exponents for the spin glass phase transition for the different values
of the strength of the isotropic disorder, discussing the cross-over regime.
