Following the approach of Rota and Taylor \cite{SIAM}, we present an
innovative theory of Sheffer sequences in which the main properties are encoded
by using umbrae. This syntax allows us noteworthy computational simplifications
and conceptual clarifications in many results involving Sheffer sequences. To
give an indication of the effectiveness of the theory, we describe applications
to the well-known connection constants problem, to Lagrange inversion formula
and to solving some recurrence relations
