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Let $F_n$ be the free group on $n$ generators. Consider the group $IA_n$ of automorpisms of $F_n$ acting trivially on its abelianization. There are two canonical filtrations on $IA_n$: the first one is its lower central series $\Gamma_*$; the second one is the Andreadakis filtration $\mathcal A_*$, defined from the action on $F_n$. Andreadakis asked if and how these filtrations were different. We begin by describing a framework adapted to the study of such filtrations and their counterparts on group algebras. We then study several versions of the problem. In particular, we look at its restriction to some subgroups of $IA_n$ : we show that the two filtration coïncide when restricted to the triangular subroups and to braid groups. We also consider a stable version of the problem : we establish that the canonical morphism between the associated graded Lie rings is surjective when $n$ is big enough compared to a fixed degree. We also investigate a $p$-restricted version of the Andreadakis problem, and provide a calculation of the Lie algebra of the classical congruence group.Our methods are algebraic in nature. The tools come from combinatorial group theory and the study of mapping class groups; we often introduce some categorical langage to reformulate them.