For a group or semigroup or ring G, solving equations where the coefficients are elements in G and the solutions take values in G can be seen as akin to solving systems of linear equations in linear algebra, Diophantine equations in number theory, or more generally, polynomial systems in algebraic geometry. In this talk I will give a survey about solving equations in infinite non-abelian groups, with emphasis on free groups, and show how imposing certain constraints on the solutions can tilt the balance between decidability and undecidability.