Colm-cille Caulfield University of Cambridge This lecture aims to describe some interesting challenges in scalar mixing in turbulent incompressible flows, particularly focussing on flows of environmental and industrial/engineering relevance. A key aim for the applied mathematical fluid dynamics community is to gain sufficient physical and mathematical insight to develop parameterisations that can then be used by scientists and engineers to model turbulent mixing within large scale complex flows. In such flows, time-dependence, domain geometry, anisotropy, and non-newtonian fluid properties can all be of central importance, making very high demands on the required quality of a parameterisation. A further, and very important, complicating factor is when the scalar that is being mixed is `dynamic’ (e.g. temperature, salinity or density) as mixing a dynamic scalar repartitions energy between the kinetic, internal and gravitational potential energy reservoirs in subtle, and still imperfectly understood ways. Some of the open problems which will be discussed in this lecture include: 1) Measures of mixing. What are appropriate ways to define `efficiency’ in mixing, particularly in flows subject to constraints? What are the connections between idealised flows and irreversible mixing in flows with finite diffusivities? Can useful distinctions be drawn between larger scale `stirring’ and smaller scale irreversible `mixing’? 2) Buoyancy effects. How do density variations modify mixing? In particular, how does a larger scale statically stable density distribution affects mixing? Is mixing in a stratified fluid appropriately modelled as a diffusive process? What is the role of layering? 3) Real fluid properties. Are the convenient Boussinesq approximation and idealised equations of state fundamentally flawed descriptions of real fluids? How should `mixing' be modelled in multi-component, multiphase flows? How should mixing of non-newtonian fluids be quantified?