  Let $\mathcal{G}(4,2)$ be the set of connected regular graphs with four
distinct eigenvalues in which exactly two eigenvalues are simple,
$\mathcal{G}(4,2,-1)$ (resp. $\mathcal{G}(4,2,0)$) the set of graphs belonging
to $\mathcal{G}(4,2)$ with $-1$ (resp. $0$) as an eigenvalue, and
$\mathcal{G}(4,\geq -1)$ the set of connected regular graphs with four distinct
eigenvalues and second least eigenvalue not less than $-1$. In this paper, we
prove the non-existence of connected graphs having four distinct eigenvalues in
which at least three eigenvalues are simple, and determine all the graphs in
$\mathcal{G}(4,2,-1)$. As a by-product of this work, we characterize all the
graphs belonging to $\mathcal{G}(4,\geq-1)$ and $\mathcal{G}(4,2,0)$,
respectively, and show that all these graphs are determined by their spectra.
