The Jacobi-Perron algorithm is a natural way of generalising the continued fractions algorithm in the $d$-dimensional case. As in the one-dimensional case, if we denote by $T$ the associated map on the $d$-dimensional torus, there exists a unique $T$-invariant probability measure $\mu $ which is absolutely continuous with respect to the Lebesgue measure. Its density is a strictly positive function. The quality of the Diophantine approximations obtained by this algorithm is given by some inequalities between the characteristic exponents $\lambda _{1},\lambda _{2}, \ldots , \lambda _{d+1}$ of a product of matrices related to $T$. In this survey, we define these exponents, we show how they provide information on the quality of the approximations obtained by this algorithm and we give some explanations and an outline of the proof of the inequalities $ \lambda _{1}>\lambda _{2}>\cdots >\lambda _{d}>\lambda _{d+1} \mbox { and } \lambda _{2}+\lambda _{3}+\cdots +\lambda _{d}<0, $ which hold $\mu $-almost everywhere. At the end of the paper we construct, in the 2-dimensional case, a set of points for which the exponents exist and satisfy $\lambda _{2}=\lambda _{3}$. This set is dense in the square and $T$-invariant. For these points, the Jacobi-Perron algorithm gives some good simultaneous approximations in the Dirichlet sense.