AUTHOR=Brum Javier , Benech Nicolás , Gallot Thomas , Negreira Carlos TITLE=Shear Wave Elastography Based on Noise Correlation and Time Reversal JOURNAL=Frontiers in Physics VOLUME=Volume 9 - 2021 YEAR=2021 URL=https://www.frontiersin.org/journals/physics/articles/10.3389/fphy.2021.617445 DOI=10.3389/fphy.2021.617445 ISSN=2296-424X ABSTRACT=Shear wave elastography (SWE) relies on the generation and tracking of coherent shear waves. Recent technological developments have allowed SWE to be implemented in commercial ultrasound and magnetic resonance imaging systems, quickly becoming a new imaging modality in medicine and biology. However, coherent shear wave tracking sets a limitation to SWE because it either requires ultrafast frame rates (of up to 20 kHz), or alternatively, a phase-lock synchronization between shear wave-source and imaging device.} Moreover, there are many applications where coherent shear wave tracking is not possible because \textcolor{red}{scattered waves from tissue's inhomogeneities or waves coming from muscular activity, heart beating or external vibrations interfere with the coherent shear wave. To overcome these limitations, several authors developed an alternative approach to extract the shear elasticity of tissues from a complex elastic wavefield. To control the wavefield, this approach relies in the analogy between time reversal and seismic noise cross correlation. By cross-correlating the elastic field at different positions, which can be interpreted as a time reversal experiment performed in the computer, shear waves are virtually focused on any point of the imaging plane. Then, different independent \textcolor{red}{methods} can be used to image the shear modulus, for example, tracking the coherent shear wave as it focuses, measuring the focus size or simply evaluating the amplitude at the focusing point. The main advantage of this approach is its compatibility with low imaging rates modalities, which has led to innovative developments and new challenges in the field of multi-modality elastography. The goal of this short review is to cover the major developments in wave-physics involving elasticity imaging using a complex elastic wavefield and its latest applications including slow imaging rate modalities and passive elasticity imaging based on physiological noise correlation.