A plethora of natural and engineering flows exhibit spatial and temporal variation in viscosity. The variations occur in flows across a colossal range of length and time scales, from microbial motion to Earth-scale mantle convection, and give rise to new physical mechanisms that are absent in constant-viscosity fluid flows. This review surveys such phenomena across scales, and examines these mechanisms. In Stokes (zero Reynolds number) flows, viscosity gradients lead to translation-rotation coupling, and enable novel particle dynamics such as rotation in response to uniform forcing – a feature that may be exploited by microorganisms. The dynamics of laminar shear flows across scales are transformed by viscosity variation, by modifications to the base flow profile, and by the breaking of symmetries. In high-Reynolds-number shear flow, viscosity stratification fundamentally alters the mathematical structure of the singular perturbation problem which describes the production of disturbance kinetic energy. Even a minor stratification within the layer of kinetic energy production can dramatically enhance or suppress standard instabilities, and create new instabilities. Shear flows are prone to transition to turbulence via linear mechanisms: here the linear stability operator is non-normal, resulting in algebraic perturbation growth. Viscosity variations introduce new non-normal and nonlinear pathways for perturbation energy growth, apart from altering the structure and dynamics of the perturbations by the broken symmetries. While laminar and fully developed turbulent flows have received enormous attention over the decades, the process by which laminar flows transition to turbulence is not well-understood except in a few canonical constant-viscosity flows, and there too, only partially. There are myriad routes to fully-developed turbulence, and viscosity-variations will likely occupy centre-stage as we learn more about them. In fully developed turbulence, viscosity variations influence the structure of wall-bounded flows, jets, and mixing layers. At even larger scales, accounting correctly for the stratification of eddy viscosity in the ocean in global circulation models can lead to better weather predictions. Moving to the planetary scale, viscosity variations of several orders of magnitude occur in Earth’s mantle, and these play a central role in geological evolution and mantle convection. Flows laden with solid particles and larger objects, whether in dilute suspension or in mushy regimes, are ubiquitous across scales. Effective viscosity variations from inhomogeneous particle loading can produce effects similar to actual viscosity variations. Throughout this review, we emphasize scenarios where viscosity variation qualitatively, and not merely quantitatively, alters flow physics. Although some aspects of viscosity varying flows have been studied for decades, an understanding of the physics of many other aspects is just beginning, we believe, with big questions lying wide open. The review is written with graduate students in mind, and in every discussion we attempt to identify well-posed, tractable research questions. We highlight cross-scale, and therefore cross-disciplinary, connections, which are rarely made but will likely be insightful.