We introduce a reaction-diffusion model to describe the dynamics of a biological population facing a climate shift. In this context, global warming is interpreted as a Northward shift of the favourable zone for a given species. In terms of the various parameters, the question is to know whether the species is able to keep track with the climate change and how the latter affects the population size and distribution. The reaction-diffusion model will be seen to shed a new light on these questions. This is joint work with Odo Diekman, Kees Nagelkerke and Paul Zegeling (Bull. Math. Biol. 2009). Mathematically, one is led to questions of travelling fronts with forced speeds. In the second part of my talk, I will present extensions to higher dimensions and to more general geometries, reporting on joint works with Luca Rossi. These extensions involve nonlinear elliptic and parabolic equations in unbounded domains and generalized principal eigenvalues of elliptic and parabolic operators. I will discuss some of the mathematical issues and recent results for these questions.