  Let $G$ be a finite group, $S\subseteq G\setminus\{1\}$ be a set such that if
$a\in S$, then $a^{-1}\in S$, where $1$ denotes the identity element of $G$.
The undirected Cayley graph $Cay(G,S)$ of $G$ over the set $S$ is the graph
whose vertex set is $G$ and two vertices $a$ and $b$ are adjacent whenever
$ab^{-1}\in S$. The adjacency spectrum of a graph is the multiset of all
eigenvalues of the adjacency matrix of the graph. A graph is called integral
whenever all adjacency spectrum elements are integers. Following Klotz and
Sander, we call a group $G$ Cayley integral whenever all undirected Cayley
graphs over $G$ are integral. Finite abelian Cayley integral groups are
classified by Klotz and Sander as finite abelian groups of exponent dividing
$4$ or $6$. Klotz and Sander have proposed the determination of all non-abelian
Cayley integral groups. In this paper we complete the classification of finite
Cayley integral groups by proving that finite non-abelian Cayley integral
groups are the symmetric group $S_{3}$ of degree $3$, $C_{3} \rtimes C_{4}$ and
$Q_{8}\times C_{2}^{n}$ for some integer $n\geq 0$, where $Q_8$ is the
quaternion group of order $8$.
