This chapter introduces the umbral calculus, a linear-algebraic theory used to study certain polynomial functions important in applied mathematics. The focus is on the algebraic aspects rather than applications. The lower factorial numbers are defined as (n)k = n(n-1)...(n-k+1). Formal power series are discussed, with F denoting the algebra of formal power series in variable t with complex coefficients. A formal power series is a sum of the form f(t) = Σa_k t^k. Addition and multiplication are formal operations. The order o(f) of a series f is the smallest exponent of t with a nonzero coefficient. A series has a multiplicative inverse if and only if its order is 0. It is shown that o(fg) = o(f) + o(g) and o(f+g) ≥ min{o(f), o(g)}. If o(f_k) → ∞ as k → ∞, then substituting f_k for t^k in a series g(t) yields a well-defined series h(t). If o(f) ≥ 1, then the composition g(f(t)) is well-defined, and o(g∘f) = o(g)o(f). If o(f) = 1, then f has a compositional inverse, called a delta series. The powers of a delta series form a pseudobasis for F, meaning any series g ∈ F can be uniquely expressed as a sum of a_k f^k(t). The formal derivative of a series is given by ∂_t f(t) = Σk a_k t^{k-1}, and ∂_t is a derivation, satisfying ∂_t(fg) = ∂_t(f)g + f∂_t(g).This chapter introduces the umbral calculus, a linear-algebraic theory used to study certain polynomial functions important in applied mathematics. The focus is on the algebraic aspects rather than applications. The lower factorial numbers are defined as (n)k = n(n-1)...(n-k+1). Formal power series are discussed, with F denoting the algebra of formal power series in variable t with complex coefficients. A formal power series is a sum of the form f(t) = Σa_k t^k. Addition and multiplication are formal operations. The order o(f) of a series f is the smallest exponent of t with a nonzero coefficient. A series has a multiplicative inverse if and only if its order is 0. It is shown that o(fg) = o(f) + o(g) and o(f+g) ≥ min{o(f), o(g)}. If o(f_k) → ∞ as k → ∞, then substituting f_k for t^k in a series g(t) yields a well-defined series h(t). If o(f) ≥ 1, then the composition g(f(t)) is well-defined, and o(g∘f) = o(g)o(f). If o(f) = 1, then f has a compositional inverse, called a delta series. The powers of a delta series form a pseudobasis for F, meaning any series g ∈ F can be uniquely expressed as a sum of a_k f^k(t). The formal derivative of a series is given by ∂_t f(t) = Σk a_k t^{k-1}, and ∂_t is a derivation, satisfying ∂_t(fg) = ∂_t(f)g + f∂_t(g).