Let $G$ be a group. Consider an arbitrary equation given by $w(\vec{g})=e$, where $w: G^n \to G$ takes an $n$-tuple $(g\_1,...,g\_n)$ to some expression involving products of the $g\_i$, their inverses and other elements of $G$.


Are there any good methods to determine whether such equation has a solution $\vec{g} \in G^n$ or not? For which classes of groups do they work? I'd appreciate references on related results. 



Results about the existence of solutions in groups
If you are interested in finite simple groups, there is a whole raft of literature considering the related question of which words maps $w: G^n\to G$ are **surjective**. The answer is often yes, so this gives a strong affirmative answer to your question in this case.


In general you should search the literature for work by Shalev, Garion, Liebeck, O'Brien, Larson, Tiep (and many others; please forgive me as I know I've missed many people out!)


Specific references:


* M. W. Liebeck and A. Shalev, *Diameters of finite simple groups: sharp bounds and applications*, Ann. of Math. 154 (2001) 383–406.


The main result here is that for a given non-trivial word $w$, every
element of every sufficiently large finite simple group $G$ can be expressed as a product of $C(w)$ values of $w$ in $G$, where $C(w)$ depends only on $w$;
* M. Larsen and A. Shalev, *Word maps and Waring type problems*, J. Amer. Math. Soc. 22 (2009) 437–466.


M. Larsen, A. Shalev and P. H. Tiep, *The Waring problem for finite simple groups*, Ann. of Math. 174 (2011) 1885–1950.


A. Shalev, *Word maps, conjugacy classes, and a noncommutative Waring-type theorem*, Ann. of Math. 170 (2009) 1383–1416.


These three papers improve the $C(w)$ from the last bullet point to $2$. This is best possible - improving C(w) to 1 is not possible in general, as is shown by power words $x^n$ , which cannot be surjective on any finite group of order non-coprime to $n$.
* There are a host of other results in the same direction, so I can't be
exhaustive. I'll just mention one little paper I came across recently
which is related and which I thought was pretty groovy.


Sebastian Jambor, Martin W. Liebeck and E. A. O’Brien, *Some word maps that are non-surjective on infinitely many
finite simple groups,*
Bull. London Math. Soc. 45 (2013) 907–910


