The methods used by modern topology are very diverse. This book discusses in detail the methods of combinatorial topology, which consist in the study of topological spaces through their partition into some elementary sets , and methods of differential topology , which consist in the consideration of smooth manifolds and smooth mappings. Often the same topological problem can be solved both by combinatorial methods and differential. In such cases, both approaches are discussed. One of the main goals of the book is to advance in the study of the properties of topological spaces (and especially varieties) as far as possible without involving complex techniques. This is different from most topology books. The book contains a lot of tasks and exercises. Almost all tasks are equipped with detailed solutions.