This chapter starts with a number of samples of optimization problems and introduces the formal definition for optimization problems. Necessary background in computer algorithms will be reviewed. We then discuss in detail two important sample optimization problems: the MINIMUM SPANNING Tree problem and the Matrix-Chain Multiplication problem. Algorithms and analysis are given for the problems. Two basic techniques, the greedy method and the dynamic programming method, are illustrated in the study of these two problems. Finally, we give a brief discussion on NP-completeness theory, which will play an important role throughout the book.