The book is a comprehensive treatment on various topological and geometrical aspects of the theory of loops (i.e., generally non-associative binary structures with neutral elements and unique division). The loops are represented as stable transversals (alias sections) in their one-sided multiplication groups, and these are equipped with additional structure. Thus topological, differentiable, algebraic loops are treated via Lie groups, algebraic groups, symmetric spaces, etc. The book is divided into two parts. The first one presents basic material, with special attention paid to Bol loops and Bruck loops. The second part is devoted to smooth loops on low dimensional manifolds.