The umbral transfer-matrix method. I: Foundations. In this paper the author lays the foundation for the umbral transfer-matrix method based on G. C. Rota’s realization of an umbra, merely a linear functional on a vector space of formal power series. It appears to be the first in a series of papers to be written by the author aiming to show how Rota’s concept blended with the transfer-matrix method could be gainfully employed to compute generating functions for many difficult problems dealing with counting combinatorial objects.