  This paper presents some finiteness results for the number of boundary slopes
of immersed essential surfaces of given genus g in a compact 3-manifold with
torus boundary. In the case of hyperbolic 3-manifolds we obtain uniform
quadratic bounds in g for the number of possible slopes, independent of the
3-manifold. We also look at some related quantities, such as how many times the
slopes of two such surfaces of specified genus can intersect.
