A new model of fracture mechanics which takes into account interfacial effects due to a curvature-dependent surface tension will be considered. This model is based on a physically valid assumption that the behavior of molecules near a surface of a material is significantly different from those in the bulk and depends on the local curvature of the material surface. The theory will be presented through three examples: a curvilinear non-interface crack, a straight interface crack and contact problems for a rigid stamp indentation into an elastic half-plane. It will be shown that the incorporation of surface effects on the crack boundary will eliminate the power and os- cillating singularities at the crack tips which are predicted by linear elastic fracture mechanics. The mechanical problems will be reduced to the systems of singular integro-differential equations. The regularization and numerical solution of these systems will be addressed and numerical examples will be presented. Potential direction for future research and connections with experimental results will be discussed.