My main interests are in modern geometry/topology, algebra and mathematical physics. I observe that there is a raising communication, language and social barrier between this community and the differential equations community, with the main exception of the study of geometrical PDEs. The areas of mathematics which are close to my knowledge have an impressive list of new appearing frameworks, which are changing the landscape and have the capacity of even redefining the basics of the subject. To mention just few which emerged in roughly last 15 years: derived algebraic geometry, tropical geometry, geometry over "the field of one element" $\mathbf{F}_1$, cluster algebras, higher categories, $(\infty,1)$-topoi, Kontsevich deformation quantization, homotopy type theory and univalent foundations, categorification in representation theory, derived categories of motives, $\mathbf{A}_1$-homotopy theory, geometric Langlands program, motivic integration, relation between motives and renormalization of QFT, geometric understanding of BV-formalism etc. It is hard to think of the subject without those, despite the fact that these framework appeared that recently.
I am interested of learning of similar recent landscape changes in PDEs (apart from the description, please take the subject in quite general sense, including variational calculus, stochastic PDEs etc. and give some pointers to the names or seminal references).
It is hard to spot them from outside, and it s hard to believe that there is nothing vaguely comparable to the advances of the previous epoch (most notably, the work of Hoermander in 1960s on Fourier integral operators and linear and quasilinear PDEs; or, say, the work of Gromov on h-principle for partial differential relations in differential geometry).
Though, there is no need of repeating, as I am well aware of: the framework of polyfolds and general Fredholm theory due Hofer, and PDEs related to symplectic and contact geometry in general; then, of course, breakthrough in the study of Ricci flow after Perelman; the appearance of systematics methods of obtaining local index formulas by Connes and others; then advances related to integrable PDEs (including, most recently, the development of methods involving nonlinear Poisson vertex algebras by Victor Kac and collaborators), and those related to microlocal analysis and hyperfunctions; to cohomological analysis of PDEs related to secondary calculus. I am aware of the importance of the works of Villani and also of Tao, though I have somewhat too vague picture of which are the new general fundamental principles there.
I am also aware that the numerical control and theoretical study of special cases of Navier-Stokes got to much higher level than before and that the homological algebra is recently systematically applied to the study of stabilty of finite element methods by Douglas Arnold and collaborators (see references). But I would like to learn of other new fundamental frameworks.
A related question is is if there are recent fundamentally new types of functional spaces which now promise to become central in the study of nonlinear PDEs.
Best Answer
A very active field of research (and to my understading, may fall into the "fundamental" category) is Domain Decomposition methods (DDM), which can be understood in the geometrical numerical and computational sense. In this last, parallel algorithms are being explored. Although many of these methods are based on Lagrange multipliers, efforts are also made to make a sensible domain decomposition without using them, through indirect methods, like Green's functions, from which some collocation methods can be derived (see "General Theory of Domain Decomposition: Indirect Methods Ismael Herrera, Robert Yates, Martin Diaz", Numerical Methods of Partial Differential Equations, Wiley). In this same paper are mentioned the Steklov-Poincare operators, which I believe is a line of research on its own right. And come to think of it, collocation methods are also a line of research.
Your related question: "[...]fundamentally new types of functional spaces[...]" You don't mention which functional spaces you are aware of, but I can mention Sobolev spaces, which support weak derivatives, and these in turn can be used in problems involving "jumps" or some type of discontinuity (for example, in combustion/explosion problems, see in this instance "physics of shock waves and High-Temperature Hydrodynamic phenomena" by Ya. B. Zeld'dovich and Yu. P. Raizer"). See also Godlewski and Raviart (1996), "Numerical approximation of hyperbolic systems of conservation laws", Applied Mathematical Sciences 118 (Springer, New York). Sobolev spaces have been used also for DDM, for example in the same paper by Herrera, Yates and Diaz.
There is this book "Navier-Stokes Equations and Turbulence" edited by C. Foias, which devotes some pages to the Banach-Tarski paradox. Maybe that famous paradox (a theorem, in fact) may provide some new avenues of research into some types of PDE's.