Turbulent flows appear in many different phenomena and is of fundamental importance in science and technology. Great part of the conventional statistical theory of turbulence, however, is based on heuristic arguments and empirical information, with the notion of ensemble average of flows playing a fundamental role. The theory of statistical solutions aims towards a rigorous foundation for the conventional statistical theory of turbulence by rigorously defining the evolution of the probability distributions of the velocity field within the framework of Leray-Hopf weak solutions of three-dimensional incompressible Navier-Stokes equations. In this talk we will review a few characteristics of turbulent flows and discuss the concept of statistical solution. We then mention some rigorous results obtained with this framework. If time permits, we discuss a generalization of the notion of statistical solution to an abstract setting that easily applies to many different equations.