A maximal inequality is an inequality which involves the (ab-solute) supremum sups\t |Xs|or the running maximum sups\t Xs of a stochastic process (Xt)t,0. We discuss maximal inequalities for several classes of stochastic processes with values in an Euclidean space: Martin-gales, Levy processes, Levy-type - including Feller processes, (compound) pseudo Poisson processes, stable-like processes and solutions to SDEs driven by a Levy process -, strong Markov processes and Gaussian processes. Us-ing the Burkholder-Davis-Gundy inequalities we also discuss some rela-tions between maximal estimates in probability and the Hardy-Littlewood maximal functions from analysis.