  The purpose of this paper is to systematically study compactness and
essential norm properties of operators on a very general class of weighted Fock
spaces over $\C$. In particular, we obtain rather strong necessary and
sufficient conditions for a wide class of operators (which includes operators
in the Toeplitz algebra generated by bounded symbols) to be compact and we
obtain related estimates on the essential norm of such operators. Finally, we
discuss interesting open problems related to our results, and in particular
discuss the possibility of extending our results to other generally weighted
Bergman spaces on the unit ball of $\C$.
