We perform high-statistics Monte Carlo simulations of three-dimensional Ising spin glass models on cubic lattices of size
L: the $\ifmmode\pm\else\textpm\fi{}J$ (Edwards-Anderson) Ising model for two values of the disorder parameter
p,
p=0.5 and
p=0.7 (up to
L=28 and
L=20, respectively), and the bond-diluted bimodal model for bond-occupation probability
pb=0.45 (up to
L=16). The finite-size behavior of the quartic cumulants at the critical point allows us to check very accurately that these models belong to the same universality class. Moreover, it allows us to estimate the scaling-correction exponent $\ensuremath{\omega}$ related to the leading irrelevant operator: $\ensuremath{\omega}=1.0(1)$. Shorter Monte Carlo simulations of the bond-diluted bimodal models at
pb=0.7 and
pb=0.35 (up to
L=10) and of the Ising spin glass model with Gaussian bond distribution (up to
L=8) also support the existence of a unique Ising spin glass universality class. A careful finite-size analysis of the Monte Carlo data which takes into account the analytic and the nonanalytic corrections to scaling allows us to obtain precise and reliable estimates of the critical exponents. We obtain $\ensuremath{\nu}=2.45(15)$ and $\ensuremath{\eta}=\ensuremath{-}0.375(10)$.