A natural question that many (researchers interested in formal languages and groups) have asked is: given a formal language class (for example regular, context-free), which finitely generated groups have word problem in that class? If $G$ is a group generated by a finite set $X$ which we can assume is inverse-closed (if $g\in X$ then $g^{-1}\in X$) and $\pi\colon X^*\to G$ sends words to the elements their product multiplies to in $G$, and $e$ is the identity element of $G$, the word problem for $(G,X)$ is the language $\{w\in X^* \mid pi(w)=e\}$. Results in this direction include: regular if and only if $G$ is finite (Anisimov); context-free if and only if $G$ is virtually free (Muller-Schupp); 1-counter if and only if $G$ is virtually cyclic (Herbst); blind multi-counter if and only if $G$ is virtually abelian (Elder, Kambites, Ostheimer) if and only if the word problem is the intersection of finitely many 1-counter languages (Holt, Owens, Thomas); and partial results include EDT0L implies $G$ is torsion (Bishop et al 2026). In my talk I will describe some attempts to fit multiple context-free (MCF) languages, introduced by Seki, Matsumura, Fujii and Kasami (1991), into the above list of results. I will define the class MCF in terms of grammars and an equivalent machine charactisation, use the machine version to prove a result which can be used to show a language is not MCF (similar to a pumping lemma) and the grammar version to prove a result about permutations of MCF languages. This is joint work with Andrew Duncan, Lisa Frenkel and Mengfan Lyu.