We study a large class of BPS surface defects in 4d N=2 gauge theories. They
are defined by coupling a 2d N=(2,2) gauged linear sigma model to the 4d bulk
degrees of freedom. Our main result is an efficient computation of the
effective twisted superpotential for all these models in terms of a basic
object closely related to the resolvent of the 4d gauge theory, which encodes
the curve describing the 4d low energy dynamics. We reproduce and extend the
results of brane constructions and compute the effective twisted superpotential
for general monodromy surface defects. We encounter novel, puzzling field
theory phenomena in the low energy dynamics of the simplest surface defects and
we propose some local models to explain them. We also study in some detail the
behavior of surface defects near monopole points of the bulk theory's Coulomb
branch. Finally, we explore the effect on the defect of breaking the bulk
supersymmetry from N=2 to N=1 and show that certain quantities are independent
of this breaking.