This review presents universal aspects of stochasticity of simple A.5- or 2-degree-of-freedom) Hamiltonian systems. Stochasticity the seemingly erratic wandering of orbits of non-integrable Hamiltonian systems over some part of phase space, accompanied by exponential
ivergence of nearby orbits. It is a large-scale phenomenon that spreads over larger and larger regions of phase space by the successive breakups of
jrriers called Kolmogorov-Arnold-Moser (KAM) tori when some perturbation to an integrable Hamiltonian is increased. The main emphasis of this review is on the breakup of KAM tori which is described by a renormalization group for Hamiltonians of the KAM type. This paper also reports :cent progress in describing chaotic transport which is the large scale manifestation of stochasticity, but this is not the last word to chaos. The central model of this paper is the Hamiltonian of one particle in two longitudinal waves, H?(v, x, t)= v2l2- M cosx- P cos k(x- t), which is a iradigm for simple Hamiltonian systems. Simple approximate renormalization schemes for KAM tori of Hp are derived, and the way to exactly normalize a general Hamiltonian of the KAM type is explained as well.