Analysis in singular spaces is becoming an increasingly important area of research, with motivation coming from the calculus of variations, PDEs, geometric analysis, metric geometry and probability theory, just to mention a few areas. In all these fields, the role of measure theory is crucial and an appropriate understanding of the interaction between the relevant measure-theoretic framework and the objects under investigation is important to a successful research. The aim of this book, which gathers contributions from leading specialists with different backgrounds, is that of creating a collection of various aspects of measure theory occurring in recent research with the hope of increasing interactions between different fields. List of contributors: Luigi Ambrosio, Vladimir I. Bogachev, Fabio Cavalletti, Guido De Philippis, Shouhei Honda, Tom Leinster, Christian Léonard, Andrea Marchese, Mark W. Meckes, Filip Rindler, Nageswari Shanmugalingam, Takashi Shioya, and Christina Sormani.