Understanding transport of thermal and suprathermal particles is a fundamental issue in laboratory, solar-terrestrial, and astrophysical plasmas. For laboratory fusion experiments, confinement of particles and energy is essential for sustaining the plasma long enough to reach burning conditions. For solar wind and magnetospheric plasmas, transport properties determine the spatial and temporal distribution of energetic particles, which can be harmful for spacecraft functioning, as well as the entry of solar wind plasma into the magnetosphere. For astrophysical plasmas, transport properties determine the efficiency of particle acceleration processes and affect observable radiative signatures. In all cases, transport depends on the interaction of thermal and suprathermal particles with the electric and magnetic fluctuations in the plasma. Understanding transport therefore requires us to understand these interactions, which encompass a wide range of scales, from magnetohydrodynamic to kinetic scales, with larger scale structures also having a role. The wealth of transport studies during recent decades has shown the existence of a variety of regimes that differ from the classical quasilinear regime. In this paper we give an overview of nonclassical plasma transport regimes, discussing theoretical approaches to superdiffusive and subdiffusive transport, wave-particle interactions at microscopic kinetic scales, the influence of coherent structures and of avalanching transport, and the results of numerical simulations and experimental data analyses. Applications to laboratory plasmas and space plasmas are discussed.