The chapter analyses the development in time of some physical system subject to random influences, by means of a discrete or continuously registering instrument. The behaviour of maxima in random sequences and stochastic processes having as a target to obtain, under convenient conditions, the same limiting distributions is discussed in this chapter. Extremes in stochastic sequences, global extremes of Gaussian stationary processes, global extremes in Wiener-Lévy processes and local extremes in Gaussian stationary processes are given with some results. Some scattered examples of extreme value properties for other kinds of stochastic processes are also discussed.