This collection of original articles and surveys, emerging from a 2011 conference in Bertinoro, Italy, addresses recent advances in linear and nonlinear aspects of the theory of partial differential equations (PDEs). Phase space analysis methods, also known as microlocal analysis, have continued to yield striking results over the past years and are now one of the main tools of investigation of PDEs. Their role in many applications to physics, including quantum and spectral theory, is equally important. Key topics addressed in this volume include: general theory of pseudodifferential operators Hardy-type inequalities linear and non-linear hyperbolic equations and systems Schrödinger equations water-wave equations Euler-Poisson systems Navier-Stokes equations