New Trends on Analysis and Geometry in Metric Spaces : Levico Terme Italy 2017 This book includes four courses on geometric measure theory the calculus of variations partial differential equations and differential geometry. Authored by leading experts in their fields the lectures present different approaches to research topics with the common background of a relevant underlying usually non Riemannian geometric structure. In particular the topics covered concern differentiation and functions of bounded variation in metric spaces Sobolev spaces and differential geometry in the so called Carnot–Carathéodory spaces. The text is based on lectures presented at the 10th School on Analysis and Geometry in Metric Spaces held in Levico Terme (TN) Italy in collaboration with the University of Trento Fondazione Bruno Kessler and CIME Italy. The book is addressed to both graduate students and researchers.