  We prove that C^1-robustly transitive diffeomorphisms on surfaces with
boundary do not exist, and we exhibit a class of diffeomorphisms of surfaces
with boundary which are C^k-robustly transitive, with k greater or equal than
2. This class of diffeomorphisms are examples where a version of Palis'
conjecture on surfaces with boundary, about homoclinic tangencies and uniform
hyperbolicity, does not hold in the C^2-topology. This follows showing that
blow-up of pseudo-Anosov diffeomorphisms on surfaces without boundary, become
C^2-robustly topologically mixing diffeomorphisms on a surfaces with boundary.
