In this chapter we study how small changes in the entries of a matrix affect the eigenvalues and eigenvectors of that matrix. Such changes occur in modeling since the matrix in the model is often only an approximation of the actual one. Further, in numerical computations, usually the answer we get is (due to rounding) actually the exact answer to a matrix which is close to the given matrix. So, if matrices close to a given matrix have close eigenvalues and eigenvectors, we would have an ideal situation. Unfortunately, this is not always the case.