Beginning with a brief introduction to algorithms and diophantine equations, this volume aims to provide a coherent account of the methods used to find all the solutions to certain diophantine equations, particularly those procedures which have been developed for use on a computer. The study is divided into three parts, the emphasis throughout being on examining approaches with a wide range of applications. The first section considers basic techniques including local methods, sieving, descent arguments and the LLL algorithm. The second section explores problems which can be solved using Baker's theory of linear forms in logarithms. The final section looks at problems associated with curves, mainly focusing on rational and integral points on elliptic curves. Each chapter concludes with a useful set of exercises. A detailed bibliography is included. This book will appeal to graduate students and research workers, with a basic knowledge of number theory, who are interested in solving diophantine equations using computational methods.