I know some theory of "classical" modular forms, that is functions in the complex upper-half plane satisfying

$f(\frac {az+b} {cz+d})=(cz+d)^kf(z)$

I know one can study modular forms on finite-index subgroups of $SL(2,\mathbb Z)$. But I have not seen much theory of modular forms on arbitrary Fuchsian groups. Which are the most interesting cases of such groups? Can somebody recommend a good reference?

I have also come across Hilbert and Siegel modular forms, but I don't have these in mind as an answer to this question. I wonder whether one can use arbitrary Lie group instead of $SL(2,\mathbb R)$, which is what the Wikipedia page about automorphic forms suggests, but I am not on a level to tackle the theory.