The conjugacy problem is a classical problem, and one can argue it even older than its formulation by Dehn in 1912. It asks « how to determine whether two elements of a group are conjugate » (in a proven algorithmic way). This takes a particularly familiar form if the group is a group of automorphisms. For instance, consider two matrices in GL(n, Z). They are automorphisms of Z^n, but also of R^n. Asking whether they are conjugate is an elementary linear algebra question if the expected conjugator is in GL(n, R). It is more difficult if it has to be in GL(n, Z). For the case n=2, the situation is very pleasant: it is neither ‘trivial’ nor difficult at all ! And this is because of hyperbolic geometry (and action on hyperbolic plane and modular tree). In this talk we will rather discuss the case of automorphisms of a (non-abelian) free group (and marginally of the Mapping Class groups of surfaces). These groups have similarities with SL(2,Z) and with SL(3, Z), even though these two groups are very different from each other. In certain families of elements, some behavior of SL(2,Z) is recognisable. This allows to solve the conjugacy problem for atoroidal automorphisms of free groups, and more generally for exponential automorphisms of free groups with no « independent » polynomial subgroup.