The aim of this volume is to present some new developments and ideas in partial differential equations and mathematical analysis, including spectral analysis and boundary value problems for PDE, harmonic analysis, inequalities, integral equations, and applications. This book is a collection of short summaries of reports from lectures delivered at Tbilisi Analysis & PDE seminars and workshops. In particular, it contains some applications and several open questions aimed at inspiring further research. The volume contains 21 research articles.