According to the classical Mackey theorem, every locally convex topological vectors space admits a finest compatible locally convex vector space topology. Replacing continuous linear functionals by continuous characters, Vilenkin introduced a nice and natural counterpart of local convexity, namely local quasi-convexity, for topological abelian groups. Here the problem of the existence of a finest local quasi-convex compatible group topology (called the Mackey topology in honor of George Mackey), raised in the nineties of the last century, is still open. We discuss the recent progress in the field and present some new results (e.g., a complete solution of the problem in the case of locally quasi convex groups of finite exponent).