The volume consists of a set of surveys on geometry in the broad sense. The goal is to present a certain number of research topics in a non-technical and appealing manner.The topics surveyed include spherical geometry, the geometry of finite-dimensional normed spaces, metric geometry (Bishop-Gromov type inequalities in Gromov-hyperbolic spaces), convexity theory and inequalities involving volumes and mixed volumes of convex bodies, 4-dimensional topology, Teichmuller spaces and mapping class groups actions, translation surfaces and their dynamics, and complex higher-dimensional geometry.Several chapters are based on lectures given by their authors to middle-advanced level students and young researchers. The whole book is intended to be an introduction to current research trends in geometry.