  Scale-invariant universal crossing probabilities are studied for critical
anisotropic systems in two dimensions. For weakly anisotropic standard
percolation in a rectangular-shaped system, Cardy's exact formula is
generalized using a length-rescaling procedure. For strongly anisotropic
systems in 1+1 dimensions, exact results are obtained for the random walk with
absorbing boundary conditions, which can be considered as a linearized
mean-field approximation for directed percolation. The bond and site directed
percolation problem is itself studied numerically via Monte Carlo simulations
on the diagonal square lattice with either free or periodic boundary
conditions. A scale-invariant critical crossing probability is still obtained,
which is a universal function of the effective aspect ratio r_eff=c r where
r=L/t^z, z is the dynamical exponent and c is a non-universal amplitude.
