This thesis examines the problem of the automated re-assembly of jigsaw puzzles. The objectives of this research are as follows: to provide a clear statement of the jigsaw puzzle re-assembly problem; to find out which solution technique is best suited to this problem; to determine the level of sensitivity of the proposed solution technique when solving different variations of this problem; and to explore solution methods for solving incomplete jigsaw puzzles (puzzles with missing pieces). The jigsaw puzzle re-assembly problem has been investigated only intermittently in the research literature. This work presents an extensive examination of the suitability and efficiency of the standard solution techniques that can be applied to this problem. A detailed comparison between different solution methods including Genetic Algorithms, Simulated Annealing, Tabu Search and Constraint Satisfaction Programming, shows that a constraint-based approach is the most efficient method of generating solutions to the jigsaw puzzle problem. The proposed re-assembly algorithm is successful. Consequently, it can be used in development of automated solution generators for other problems in the same domain, thus creating new theoretical and applied directions in this field of research. One potential theoretical line of research concerns jigsaw puzzles that do not have a complete set of puzzle pieces. These incomplete puzzles represent a difficult aspect of this problem that is outlined but can not be resolved in the current research. The computational experiments conducted in this thesis demonstrate that the proposed algorithm being optimised to re-assemble the jigsaw puzzles is not efficient when applied to the puzzles with missing pieces. Further work was undertaken to modify the proposed algorithm to enable efficient re-assembly of incomplete jigsaw puzzles. Consequently, an original heuristic strategy, termed Empty Slot Prediction, was developed to support the proposed algorithm, and proved successful when applied to certain sub-classes of this problem. The results obtained indicate that no one algorithm can be used to solve the multitude of possible scenarios involved in the re-assembly of incomplete jigsaw puzzles. Other variations of the jigsaw puzzle problem that still remain unsolved are presented as avenues for future research. The solution of this problem involves a number of procedures with significant applications in other computer-related areas such as pattern recognition, feature and shape description, boundary-matching, and heuristic modelling. It also has more practical applications in robotic vision and reconstruction of broken artefacts in archaeology.