The short introduction is followed by Chapter 2, which plays a double role. On the one hand, we present here all notions necessary for the under-standing of the later parts: the greatest non-representable number , the number of non-representable numbers , and the extremal versions of these: max and max , where On the other hand, our presentation of 15 characteristic problems is planned to serve also educational purposes. These include problems of competitions for college students and international mathematical olympics, shorter proofs of some theorems from earlier publications, representing typical methods of the topic, and also some new propositions of the author . According to our assumption and experience these problems can be used for theaching talented students. Besides the solutions of the above-mentioned problems we include historical remarks, comment on new results and mention also open questions.