We prove an optimal Gaussian upper bound for the densities of isotropic random walks on R^d in spherical case (d ≥ 2) and ball case (d ≥ 1). We deduce the strongest possible version of the Central Limit Theorem for the isotropic random walks: if S~_n denotes the normalized random walk and Y the limiting Gaussian vector, then Ef(S~_n)→Ef(Y) for all functions f integrable with respect to the law of Y. We call such result a “Strong CLT”. We apply our results to get strong hypercontractivity inequalities and strong Log-Sobolev inequalities.