  A Baxter algebra is a commutative algebra $A$ that carries a generalized
integral operator. In the first part of this paper we review past work of
Baxter, Miller, Rota and Cartier in this area and explain more recent work on
explicit constructions of free Baxter algebras that extended the constructions
of Rota and Cartier. In the second part of the paper we will use these explicit
constructions to relate Baxter algebras to Hopf algebras and give applications
of Baxter algebras to the umbral calculus in combinatorics.
