  We generalize the notion of a graph automatic group introduced by
Kharlampovich, Khoussainov and Miasnikov (arXiv:1107.3645) by replacing the
regular languages in their definition with more powerful language classes. For
a fixed language class $\mathcal C$, we call the resulting groups $\mathcal
C$-graph automatic. We prove that the class of $\mathcal C$-graph automatic
groups is closed under change of generating set, direct and free product for
certain classes $\mathcal C$. We show that for quasi-realtime counter-graph
automatic groups where normal forms have length that is linear in the geodesic
length, there is an algorithm to compute normal forms (and therefore solve the
word problem) in polynomial time. The class of quasi-realtime counter-graph
automatic groups includes all Baumslag-Solitar groups, and the free group of
countably infinite rank. Context-sensitive-graph automatic groups are shown to
be a very large class, which encompasses, for example, groups with unsolvable
conjugacy problem, the Grigorchuk group, and Thompson's groups $F,T$ and $V$.
