Plasma turbulence is a ubiquitous phenomenon, influencing the dynamics in most of the visible universe and playing a crucial role in countless experiments of basic and applied plasma science. Its comprehension and control is a prerequisite to the realization of fusion energy. At the same time, it constitutes a fascinating example of spatiotemporal chaos caused by the nonlinear coupling of many degrees of freedom in open systems. In recent years, new theoretical ideas, advanced computational methods, and improved measurement techniques have led to significant progress. Nevertheless, various fundamental aspects of turbulence in laboratory and astrophysical plasmas are still not well understood. The aim of this talk is to review the state of the art and outline future challenges in this active area of research.