Systems characterized by complex nonlinear dynamics lie at the heart of 21st century technology. Examples are turbulent flows in the transport and aviation industries, smart energy networks, and models of cell dynamics used in synthetic biology. Quantitative analysis of such systems using direct numerical simulations sometimes requires prohibitively large computational resources even when one is interested only in some average properties, such as mean power consumption, because all time and length scales across which the system evolves must be resolved. In addition, while numerical simulations offer detailed information starting from a specific initial state, they cannot provide safety-critical performance or stability guarantees that hold for all possible initial states. In this talk, I will describe an alternative approach to studying nonlinear systems with polynomial dynamics, which combines ideas from Lyapunov's stability theory with recent numerical tools for polynomial optimization. In particular, I will present a range of examples that demonstrate how this optimization-based method enables the efficient algorithmic construction of stability certificates and the computation of rigorous bounds on performance-related system properties. Other applications, including optimal control and disturbance amplification analysis, will be discussed along with open problems and future research directions.