In this thesis, it is explicitly shown that the exact Green's function and the unknown boundary variables on the boundary, in a given boundary value problem (BVP), satisfy the same boundary integral equation (BIE) but with a different known vector. Indeed, it is made explicit here that in using the BIE method to solve a given boundary value problem, one has in fact constructed the Green's function for the domain. This observation provides a way to construct a library of numerical approximations to exact Green's functions (discretized Green's functions) for problems for which analytical Green's functions are not available. This library thus can be used repeatedly by non-experts;The main ingredient of a discretized Green's function, for a given BVP involving one or two separate surfaces, is identified and implemented for specific applications. Some efficient strategies are proposed;In this thesis, two specific classes of problems are considered as applications of the BEM and the discretized Green's function library. One is the application of the BEM to the analysis of 2D micromechanical behavior of fiber-reinforced composites. A BEM model is developed based on models for both perfectly-bonded and imperfectly-bonded materials in a unit cell. The idea of a library of Green's functions and the entries for the library for fiber-reinforced composites are discussed;The other class of problems considered here involves elastodynamic frequency-dependent wave motion in a halfspace. Radiation from a void inside the halfspace and the scattering from a halfspace surface-breaking crack are considered using a conventional BIE (CBIE) and a hypersingular BIE (HBIE) formulations, respectively. Some new insight into this class of problems was gained during the research. As a result, strategies are suggested to exploit the best features of the fullspace Stokes and halfspace Lamb solutions. A 'parallel' scheme is also designed and implemented when Lamb's solution is used in the BIE formulation. The partitioning method, which is closely related to the process of creating and using a region-dependent Green's function, is also implemented and the efficiency of the Green's function library idea is demonstrated.