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INTRODUCTION This short review covers a wide selection of topics from a multidisciplinary area of dynamics of nonequilibrium systems in physics, chemistry, biology. Theoretical models of colloid particle and protein deposition and adhesion at surfaces, accompanied by relaxation processes, of reaction kinetics with and without diffusion, of phase coarsening and nucleation, will be surveyed. The unifying feature of these systems is the importance of fluctuations and many-body, multiparticle collective effects, in determining the dynamical behavior which is typically nonclassical. Recently a range of methods has been employed to study models of these phenomena , including exact solutions, analytical and asymptotic methods, and large-scale numerical simulations. The review is largely devoted to summary of several recent advances and provides a guide to the literature. No detailed derivations are given; our aim is to overview the field emphasizing the unifying aspects of various phenomena surveyed and list some open research directions. Most actual applications of the methods of statistical physics to practical problems as well as to scientific experiment interpretation involve time-dependent processes. Generally , nonequilibrium statistical physics has developed slower than equilibrium theories. However, recently there has been a resurgence of interest and activity, as well as reports of new advances, in the field of nonequilibrium statistical mechanical systems. The reason for this recent progress has been several-fold. Firstly, guided by experience with the equilibrium case, attention was focussed on lattice models with stochastic dynamics, rather than on direct examination of phenomenological, continuum evolution equations popular in " traditional " studies of reaction-diffusion and nucleation phenomena. Secondly, emphasis has been put on identifying those regimes where the system's behavior is dominated by nonclassical, non-mean-field fluctuations. This has lead to consideration of rather low dimensions, one (chains) and two (surfaces).