Abstract: Many distant high-energy astrophysical systems (such as pulsar wind nebulae, black-hole accretion flows, and jets from active galactic nuclei) contain collisionless plasmas that are relativistic, radiative, and highly nonthermal. Understanding the nature of turbulence in this extreme plasma physical regime and its implications for observations is an outstanding challenge in plasma astrophysics. Particle-in-cell (PIC) simulations have recently opened this topic to detailed, first-principles numerical and theoretical scrutiny. I will describe the latest progress on understanding relativistic kinetic turbulence. PIC simulations have demonstrated that relativistic turbulence is an efficient particle accelerator, joining the ranks of shocks and magnetic reconnection as a viable source of high-energy particles (and thus broadband radiation and cosmic rays). These simulations are now giving long-awaited tests for a line of analytic theories of stochastic particle acceleration originating with Enrico Fermi in 1949. Relativistic PIC simulations are also giving new insights into two-temperature electron-ion plasmas and radiative turbulence. The next several years promise to bring new breakthroughs into these problems.