We study the minimal displacement (Xn) of branching random walk with non-negative steps. It is shown that (Xn−EXn) is tight under a mild moment condition on the displacements. For supercritical B.R.W. (Xn) converges almost surely. For critical B.R.W. we determine the possible limit points of (Xn−EXn), and we prove a generalization of Kolmogorov's theorem on the extinction probability of a critical branching process. Finally we generalize Bramson's results on the almost sure convergence ofXn log 2/log logn.