  We consider a unit speed curve $\alpha$ in Euclidean four-dimensional space
$E^4$ and denote the Frenet frame by $\{T,N,B_1,B_2\}$. We say that $\alpha$ is
a slant helix if its principal normal vector $N$ makes a constant angle with a
fixed direction $U$. In this work we give different characterizations of such
curves in terms of their curvatures.
