This papers aims to unite two mathematical areas championed by G.-C. Rota: umbral calculus and Baxter algebras. The umbral calculus refers to algebraic combinatorics of sequences of polynomials of binomial type and other related sequences. Rota showed how this is encapsulated by a certain `umbral algebra' consisting of linear functionals on these polynomials. Baxter algebras are associative algebras equipped with an operator satisfying a relation having a parameter \(\lambda\). In the first major result of this paper, Guo shows that the umbral algebra is a free Baxter algebra with parameter \(\lambda=0\). Guo then uses this as a \textit{point de départ} to define the notion of a \(\lambda\)-umbral calculus, using the free Baxter algebra with parameter \(\lambda\). This gives rise to the construction of sequences of \(\lambda\)-binomial type. The nature of the subject requires there to be many proofs based upon manipulations of sums involving binomial coefficients. Unfortunately not all are necessary, which obscures some of the paper's intent. For example, Definition 2.1 on page 410 says that a sequence \(\{p_n(x)\}\) of power series in \(x\) has \(\lambda\)-binomial type if \[ p_n(x+y)= \sum_{k=0}^n\lambda^k\sum_{i=0}^n {n \choose i}{n\choose k} p_{n+k-i}(x)p_i(y) . \] Since \(x+y=y+x\), the variables \(x\) and \(y\) may be interchanged on the right hand side, namely \[ p_n(x+y)= \sum_{k=0}^n\lambda^k\sum_{i=0}^n {n \choose i}{n\choose k} p_i(x)p_{n+k-i}(y) . \] This proves Lemma 2.1, which asserts that \(\{p_n(x)\}\) has \(\lambda\)-binomial type if and only if this second equality holds. Thus the journal page of manipulations devoted to its proof was unnecessary.