This book is devoted to some interrelated problems of a new, rapidly developing branch of mathematics called combinatorial geometry. Common to all the problems examined here is the notion of “ cutting” a geometric figure into several “smaller pieces.” There are several different criteria for what constitutes a “ smaller piece”; hence this book necessarily treats several different problems. All the theorems proved here are very recent; the oldest of them was proved by the Polish mathematician Karol Borsuk about forty years ago. This theorem of Borsuk is the core around which all of the subsequent exposition unfolds. The most recent theorem is barely a year old. The topics treated in this book are well within the grasp of bright and interested high school students. At the same time, the book intro­ duces the reader to a number of the unsolved problems of geometry. This family of problems is the subject of another book by the same authors. Theorems and Problems in Combinatorial Geometry (Nauka, 1965). That book, however, deals chiefly with problems of three- dimensional and higher-dimensional spaces. The present book concerns itself only with problems of plane geometry, and can thus be used by high school mathematics clubs. Theorems and Problems in Combinatorial Geometry will be useful, however, to readers interested in continuing further. The remarks at the end of the book are intended for the more advanced reader.