Changes in biomechanical properties of biological soft tissues are often associated with physiological dysfunctions. Since biological soft tissues are hydrated, viscoelasticity is likely suitable to represent its solid-like behavior using elasticity and fluid-like behavior using viscosity. Shear wave elastography is a non-invasive imaging technology invented for clinical applications that has shown promise to characterize various tissue viscoelasticity. It is based on measuring and analyzing velocities and attenuations of propagated shear waves. In this review, principles and technical developments of shear wave elastography for viscoelasticity characterization from organ to cellular levels are presented, and different imaging modalities used to track shear wave propagation are described. At a macroscopic scale, techniques for inducing shear waves using an external mechanical vibration, an acoustic radiation pressure or a Lorentz force are reviewed along with imaging approaches proposed to track shear wave propagation, namely ultrasound, magnetic resonance, optical, and photoacoustic means. Then, approaches for theoretical modeling and tracking of shear waves are detailed. Following it, some examples of applications to characterize the viscoelasticity of various organs are given. At a microscopic scale, a novel cellular shear wave elastography method using an external vibration and optical microscopy is illustrated. Finally, current limitations and future directions in shear wave elastography are presented.