In this talk I will discuss the multiple layer potential approach for the treatment of boundary value problems associated with higher-order, constant coefficient, elliptic differential operators on smooth and Lipschitz domains. This study falls within the scope of the program outlined by A.P. Calderòn in his 1978 ICM plenary address in which he advocates the use of layer potentials "for much more general elliptic systems [than the Laplacian]."