We present here the English version of the Romanian first edition (V. Berinde: Ex plorare, investigare si descoperire in matematica, Editura Efemeride, Baia Mare, 2001). There are no major changes. Only a few printing errors were corrected. When transcribing Romanian names or denominations we did not use the diacrit ical marks. Our purpose is to provide an introduction to creative problem solving tech niques with particular emphasis on how to develop inventive skills in students. We present an array of 24 carefully selected themes that range over all the main chapters in elementary mathematics: arithmetic, algebra, geometry, analysis as well as applied mathematics. Main goal is to offer a systematic illustration of how to organize the natural transition from problem solving activity toward exploring, investigating and discovering new facts and results. The book is addressed mainly to students, young mathematicians, and teach ers, involved or/and actively working in mathematics competitions and training gifted people. It collects many valuable techniques for solving various classes of difficult problems and, simultaneously, offers a comprehensive introduction to cre ating new problems. The book should also be of interest to anybody who is in any way connected to mathematics or interested in the creative process and in mathematics as an art.