Dissipative partial differential equations (DPDEs) are ubiquitous in the description of real world phenomena. Some celebrated examples are the Navier-Stokes equations and the complex Ginzburg-Landau equation. Functional analysis methods such as interpolation inequalities (for example, the famous Gagliardo-Nirenberg inequality) are being used to tackle the most important features of the solutions of DPDEs, such as global existence, stability, instability and turbulence. Understanding length scales in a given DPDE allows us to gain insight into the spatio-temporal evolution of their solutions. Among the challenges for the study of DPDEs, perhaps the most famous one is: does the three-dimensional Navier-Stokes flow develop singularities in finite time? This question has attracted one million dollars as one of the challenges for this current millennium.