In general the theory and the construction of the solutions of boundary value problems is more difficult than those of initial value problems. Therefore, we begin this chapter by providing the necessary and sufficient conditions for the existence and uniqueness of the solutions of linear boundary value problems. For these problems explidt representations of the solutions arc given in terms of Green's matrkcs. For the construction of the solutions we have induded several algorithms whieh have been proposed recently. Although all these algorithms are the same in nature, namely convert the given boundary value problem to its equivalent initial value problem, in actual construction of the solutions one shows superiority over the others for which sometimes reasons can be explained. Most of these algorithms have been illustrated by solving discrete two point boundary value problems some of which are known to be unstable. The minimal solution of the difference equations which plays an important role in several branches of numerical analysis is introduced. For the construction of minimal solution classical algorithms of Miller and Olver are discussed.