Abstract: This article presents a combined experimental and theoretical overview of positronium collisions with atoms and molecules. It starts with a description of the basic properties of positronium. This is followed by a survey of experimental methods. These include positron annihilation lifetime spectroscopy (PALS and SSPALS), angular correlation of annihilation radiation (ACAR), Doppler broadening (DB), positronium beam experiments, and many-positron techniques. Theoretical developments are next described, including the Born approximation, coupled-state methods, variational techniques, stabilization methods, model potentials, Monte Carlo methods, the impulse approximation, and resonances. There then follows a critical comparison between experiment and theory on low-energy momentum transfer cross-sections, total cross-sections, integral and differential ionization cross-sections, and positronium-positronium scattering. Comparison is also made with equivelocity free electron and positron scattering, highlighting the remarkable result that positronium scattering tends to follow that of a bare electron. The article concludes with a look towards future directions.