A commutative ring with unit is called a chain ring if all its ideals form a chain under inclusion. All finite chain rings can be obtained in the following way: Let \(p\) be a prime, \(n,r>0\), \(f\in \mathbb{Z}_{p^n}[X]\) a monic polynomial, \(\deg(f)=r\) whose image in \(\mathbb{Z}_p[X]\) is irreducible and let \(\text{GR} (p^n,r): =\mathbb{Z}_{p^n}[X]/(f)\). Every finite chain ring is of the form \(\text{GR}(p^n,r) [X]/(g,p^{n-1}X^t)\), where \(g(X)=X^k-p (a_{k-1}X^{k-1} +\cdots +a_0)\), \(a_0\) unit, and moreover \(t=k\) when \(n=1\) and \(1\leq t\leq k\) when \(n\geq 2\). If \(g(X)=X^k-pa_0\) the ring is called pure. -- The first result of the paper is a classification up to isomorphism of finite pure chain rings with fixed \(p,n,r,k,t\) when \(n=2\) or when \(p\mid k\) but \(p^2\nmid k\) and \((p-1)\nmid k\). The second result is the structure of the group of units of a finite chain ring with fixed \(p,n,r,k,t\) if \((p-1)\mid k\).