Solitons, monopoles, and duality: from sine-Gordon to Seiberg-Witten

An elementary introduction into the Seiberg-Witten theory is given. Every effort is made to get it as pedagogical as possible, and within a reasonable size. The selection of the relevant material is heavily oriented towards graduate students. The basic ideas about solitons, monopoles, supersymmetry and duality are reviewed from the first principles, and they are illustrated on the simplest examples. The exact Seiberg-Witten solution to the low-energy effective action of the four-dimensional N = 2 supersymmetric pure Yang-Mills theory with the gauge group SU(2) is the main subject of the review. A generalization of that approach to other gauge groups is also discussed in some detail. Some related issues (adding matter, confinement, string dualities) are outlined.