So far, the non-linear evolution of cosmic structures is accessible only for large-scale numerical simulations. The conventional analytic treatment of cosmic density fluctuations with the hydrodynamical equations runs into severe conceptual and technical difficulties even on moderately non-linear scales. As an alternative, we have developed a microscopic, non-equilibrium, statistical theory for cosmic structure formation which avoids these difficulties by construction and allows to enter deeply into the non-linear regime of cosmic density fluctuations. The theory also allows to unify approaches to kinetic theory and hydrodynamics, offering a joint treatment of dark and baryonic matter. I will motivate and introduce this theory in simple terms, show some results on non-linear cosmic structure formation obtained so far, and discuss some possible future applications.