Consider a stochastic process \(\{\xi\) (t), \(t\geq 0\}\), stationary with marginal distribution function G. A method is given to evaluate asymptotically Prob\(\{\sup_{0\leq t\leq h}\xi (t)>u\}\) where \(u\to w^- \), w being the right end point of G. The method can be used when G belongs to one of the domains of attraction of the extremes. It is also shown that the distribution function of the sup also belongs to the same domain of attraction. Finally, applications are made to multidimensional Gaussian processes, and to extremes of Rayleigh processes. The Poisson character of \(\epsilon\)-upcrossings and local \(\epsilon\)-maxima is established.