Fundamentals of two-parameter elastic and elastic-plastic fracture mechanics are discussed from a view point of parameters of in-plane and out-of-plane crack-tip constraint. The non-singular T-stress components and the crack-tip constraint parameter A are incorporated in basic equations of elastic and elastic-plastic fracture mechanics, respectively. Models and criteria of two-parameter fracture mechanics are presented for various loading conditions in the case of solids with cracks and notches. Load separation method in elastic-plastic fracture mechanics are employed for measuring the fracture toughness of non-standard specimens with cracks or notches. An application of two-parameter fracture mechanics in some engineering problems is demonstrated.