  We study the finite-size behavior of two-dimensional spin-glass models. We
consider the +-J model for two different values of the probability of the
antiferromagnetic bonds and the model with Gaussian distributed couplings. The
analysis of renormalization-group invariant quantities, the overlap
susceptibility, and the two-point correlation function confirms that they
belong to the same universality class. We analyze in detail the standard
finite-size scaling limit in terms of TL^(1/nu) in the +-J model. We find that
it holds asymptotically. This result is consistent with the low-temperature
crossover scenario in which the crossover temperature, which separates the
universal high-temperature region from the discrete low-temperature regime,
scales as T_c(L) ~ L^(-theta_S) with theta_S \approx 0.5.
