This book is contributed to investigate the interactions between group theory and graph theory. In more than 60 years such interactions have greatly stimulated the development of each other, especially the theory of symmetric graphs has almost developed in parallel with the theory of permutation groups. In the study of permutation groups the information of point stabilizers of a primitive group is crucial to the structure of the primitive group. In this book we study primitive groups with soluble stabilizers, a classification is gievn. Then the outcome is used to classify certain classes of symmetric graphs, and several important classification results are obtained. This book is intended as a report of a series of researches concerning the two interactive fields, it also provides a reasonable coverage of basic concepts, ideas, and skills of permutation groups and symmetric graphs. The book is addressed primarily to researchers and postgraduate students in the fields. To read the book the reader is expected to have some basic knowledge of permutation groups, finite simple groups and graphs.