Which models of computation can be expressed through grammars?
There is a one-to-one correspondence between Chomsky Type-0 grammars and Turing machines.

This is exploied in the Thue programing language which allows you to write Turing-complete programs specified by an initial string and a set of string-rewriting rules (a semi-Thue grammar, which is equivalent to a type-0 grammar).

UPDATE:

Other than esoteric "Turing tar-pit" languages like Thue, various general purpose languages that allow the programmer to extend their own syntax can be used to perform Turing-complete computation during the parsing-compilation stage.

Languages in the Lisp family, in particular Common Lisp, are probably the most obvious examples, but also, broadly speaking, languages with static type checking that doesn't always need to halt, such as C++ with templates, Scala and Qi.