  We study a class of lattice field theories in two dimensions that includes
gauge theories. Given a two dimensional orientable surface of genus $g$, the
partition function $Z$ is defined for a triangulation consisting of $n$
triangles of area $\epsilon$. The reason these models are called
quasi-topological is that $Z$ depends on $g$, $n$ and $\epsilon$ but not on the
details of the triangulation. They are also soluble in the sense that the
computation of their partition functions can be reduced to a soluble one
dimensional problem. We show that the continuum limit is well defined if the
model approaches a topological field theory in the zero area limit, i.e.,
$\epsilon \to 0$ with finite $n$. We also show that the universality classes of
such quasi-topological lattice field theories can be easily classified.
Yang-Mills and generalized Yang-Mills theories appear as particular examples of
such continuum limits.
