This book, comprising two volumes, presents current research and original investigations in the theory of large cracks capable of propagating and initiating partial or complete fracture in deformable solids. The book covers a wide variety of problems relating to the behavior of bodies with cracks, and ranges from crack extension criteria to the solution of complex problems of fracture mechanics. Also, the authors consider critical and subcritical states of equilibrium under single, multiple, thermal and dynamic loadings in elastic, viscoelastic, elastic-plastic and piezoelectrical bodies with cracks. Experimental methods for the determination of crack-resistance characteristics of materials are exposed. The first volume, which we present to the reader, is dedicated to the main criteria and methods of elastic and elastic-plastic fracture mechanics. Using real examples, the authors show the result of applying different fracture criteria for determining the critical and permissible length of a fracture under static or cyclic loadings. The second volume, published as a separate book, contains a systematic study of special problems of fracture mechanics. The book is intended for scientists and practicing engineers working in the field of fracture mechanics and also for post-graduates and senior students specializing in solid mechanics and strength of machine parts.