  We find polynomial upper bounds on the number of isotopy classes of connected
essential surfaces embedded in many cusped 3-manifolds and their Dehn fillings.
Our bounds are universal, in the sense that we obtain the same explicit formula
for all 3-manifolds that we consider, with the formula dependent on the Euler
characteristic of the surface and similar numerical quantities encoding
topology of the ambient 3-manifold. Universal and polynomial bounds have been
obtained previously for classical alternating links in the 3-sphere and their
Dehn fillings, but only for surfaces that are closed or spanning. Here, we
consider much broader classes of 3-manifolds and all topological types of
surfaces. The 3-manifolds are called weakly generalized alternating links; they
include, for example, many links that are not classically alternating and/or do
not lie in the 3-sphere, many virtual links and toroidally alternating links.
