  This thesis is dedicated to random walks on spaces with non-positive
curvature. In particular, we study the case of group actions on CAT(0) spaces
that admit contracting elements, that is, whose properties mimic those of
loxodromic isometries in Gromov-hyperbolic spaces. In this context, we prove
several limit laws, among which the almost sure convergence to the boundary
without moment assumption, positivity of the drift and a central limit theorem.
In a second part, we study boundary maps and stationary measures on affine
buildings of type $\tilde{A}_2$, and we show that there always exists a
hyperbolic isometry for a non-elementary action by isometries on such a space.
Our approach involves the use of hyperbolic models for CAT(0) spaces, which
were constructed by H.~Petyt, D.~Spriano and A.~Zalloum, and measured boundary
theory, whose principles come from H.~Furstenberg.
