• Asymptotic models in flame propagation (Kyoto, 2004). As the name tells it, these notes describe some results on more or less complex mathematical models of flames: premixed flames, two-phase flames, flame balls.  Kyoto04
• 'Combustion and Shock waves' course within the French-Indian project FICUS. Tackles various mathematical methods useful in the investigation of flame models: hyperbolic equations, thermo-diffusive models, classical fluid mechanics equations, singular perturbations... Most videos seem (to my great relief) inaccessible, but some files are still available.
• Equations de Hamilton-Jacobi: aspects lagrangiens, comportement en temps grand, lien avec le transport optimal (Roscoff, 2007). Sorry, it is in French. Describes various aspects of the dynamical behaviour of viscosity solutions of Hamilton-Jacobi equations, starting from the very beginning of the theory, ending with convergence theorems, and with excursions into the Fathi KAMFAIBLE theory. Roscoff 1, Roscoff 2
• Dynamique des fronts de réaction-diffusion (Peyresq, 2008). Again in French. Intends to explain to physicists how one can describe the global stability of reaction-diffusion fronts for scalar equations. Very few proofs, except a simple account of the Berstycki-Nirenberg sliding arguments and how it can be extended to parabolic equations. Peyresq
• Regularity of minimal surfaces: a viscosity solutions approach (CIRM, 2009). A proof of the de Giorgi theorem on the regularity of minimal surfaces (a flat enough piece of set with minimal perimeter is smooth), based on a very remarkable paper of O. Savin, itself elaborating on former ideas of L. Caffarelli. There is, somewhere, a bluntly wrong statement (whose responsibility is entirely mine). I have decided not to correct it. Can you identify it? CIRM1,  CIRM2,  CIRM3     
• Partial differential equations with fractional diffusion (Barcelona, 2010). Describes some of the disorders, oddities and inconveniences that may occur when the standard second-order diffusion operator is replaced by a fractional one. Topics include the fractional KPP equation and free boundary problems with imposed jump on the Hölder quotient.  JISD-1, JISD-2, JISD-3
• Some PDEs with fractional diffusion (Milan, 2012). Explains, in a more or less pedestrian fashion, the one-phase nonlocal free boundary problem with imposed Hölder quotient: weak form, basic properties, regularity. Hints on the boundary behaviour of alpha-harmonic functions are given. Milan
• Invasions in periodic media: the Freidlin-Gärtner formula (Chicago, 2012). This is a minicourse given at this occasion. As the title of the conference does not indicate, the notes deal with a purely local phenomenon. But which, nevertheless, is quite suitable for a course of this sort. So, the afore-mentionned formula, for the propagation of the level sets in an inhomogeneous reaction-diffusion equation, is explained in detail. Chicago
• Propagation enhancement by a line of fast diffusion (Banff, 2014). This minicourse sums up what we have understood about what damage an active heterogeneity can do to KPP front propagation. The course notes are here. Slides showing numerical simulations are here.
• Describing biological invasions with nonlinear PDEs (Haifa, 2014). This is meant to explain the Aronson-Weinberger at master level, result in 4 hours, starting from comparison principles and sub/super-solutions; and trying to prove that the underlying equation as reasonable solutions in-between. This was the (healthy) challenge that the Getting started with PDEs team proposed to the speakers. Here is the result, together with some motivation slides.