Stratified shear flows, where the `background' velocity and density distribution vary over some characteristic length scales, are ubiquitous in the atmosphere and the ocean. At sufficiently high Reynolds number, such flows are commonly believed to play a key role in the transition to turbulence, and hence to be central to irreversible mixing of the density field. Parameterizations of such irreversible mixing within larger scale models of the ocean in particular is a major area of uncertainty, not least because there is a wide range of highly scattered and apparently inconsistent experimental and observational data. It is becoming increasingly appreciated that appropriately defined characteristic length scales of the flow are critically important to all stages of the flow's evolution, and that such data scatter is associated with differing length scales being important in different experiments and observations. Here, I review some of the recent progress using modern mathematical techniques in developing understanding of instability, transition, turbulence and mixing in stratified shear flows, focussing in particular on the crucial role of various length scales. I highlight certain non-intuitive aspects of the subtle interplay between the ostensibly stabilizing effect of stratification and destabilizing effect of velocity shear, especially when the density distribution has layers, i.e. relatively deep and well-mixed regions separated by relatively thin `interfaces' of substantially enhanced density gradient.