Abstarct: In this thesis, we investigate interpolating matrix polynomials P(\lambda) in Lagrange and Hermite baxses. A classical approach to investigating the polynomial eigenvalue problem P(\lambda)x = 0 is linearization, by which the polynomial is converted into a larger matrix pencil with the same eigenvalues. Since the current linearizations of degree n Lagrange polynomials consist of matrix pencils with n + 2 blocks, they introduce additional eigenvalues at infinity. Therefore, we introduce new linearizations which overcome this. Initially, we restrict to Lagrange and barycentric Lagrange matrix polynomials and give two new and more compact linearizations, resulting in matrix pencils of n + 1 and n blocks for polynomials of degree n . For the latter, there is a one-to-one correspondence between the eigenpairs of P(\lambda) and the eigenpairs of the pencil. We also prove that these linearizations are strong. Moreover, we demonstrate how to exploit the structure of the matrix pencils in Krylov-type methods; in this case, we only need to deal with solving linear systems of matrices with the dimension of the original matrix polynomial. Finally, we generalize for Hermite interpolation and introduce new linearizations for Hermite and barycentric Hermite matrix polynomials. Again, we can show that the linearizations are strong and that there is a one-to-one correspondence of the eigenpairs.