In this talk, we describe conditions under which maps from the euclidean space to itself  are globally injective.  In particular, we show some partial results related to the weak Markus-Yamabe conjecture, which states that if a vector field X: R^n →^Rn  has the property that, for all p in R^n, all the eigenvalues of DX(p) have negative real part, then X has at most one singularity. We will give special attention to the planar maps.