Modern concepts of fracture mechanics are presented consecutively. Homogeneous and structured models, where microstructure plays an essential role, are considered for fracture and phase transition. Firstly, one-dimensional models are comprehensively studied allowing one to retrace the main phenomena without technical difficulties. More realistic models are then used as linear and nonlinear elastic mediums, such as elastic plates with crack closure, viscoelastic discrete lattices, chains and cohesive zone models. Also considered are, crack origination, equilibrium, slow and fast growth. Sub- and super critical crack speed regimes and transition from one regime to another are studied. Fourier transform and related topics, including a version of the Wiener-Hopf technique dealing with originals are presented, as well as required topics from wave theory. This book is targeted at researchers of materials and structures, also at lecturers and advanced students.