Translated by Vladimir ShokurovThis book is actually Semester 3 of my Lectures in Geometry. It starts a new subject, however, and is there­ fore independent of the previous two semesters.The book has two major features that distinguish it from other textbooks on elementary smooth manifold the­ ory. Firstly, a lot of space is allotted to topological dimensional theory, the most geometry-oriented branch of general topology, and an acquaintance with it will bring joy to lovers of elegant mathematical constructions which provide deep insights. Secondly, I’ve ventured, in this elementary course, to present the basic notions of the theory of spectral sequences, a tool whose powerand significance is becoming increasingly clear in current studies. This is done without first expounding general (co)homology theory .The textbook ns intended as a normal course presented at two lectures a week. The number of lectures (29) arises because although the winter semester formally contains 18 weeks, in practice it is impossible to deliver more than 11 to 15 weeks of lectures. The course can be used, however, even if the curriculum assigns only one or one and a half lectures a week (11 to 15 and respectively 16 to 22 lectures).The problems in this book are mainly quite trivial and intended exclusively for a reader to test himself. Some more difficult problems are given in small print. Auxili­ ary material on algebra or calculus is also given in small print.Many thanks to the Anonymous commentor for the link to the book