  In this paper, we consider a regular curve on an oriented surface in
Euclidean 3-space with the Darboux frame $\{\mathsf{T},\mathsf{V},\mathsf{U}\}$
along the curve, where $\mathsf{T}$ is the unit tangent vector field of the
curve, $\mathsf{U}$ is the surface normal restricted to the curve and
$\mathsf{V}=\mathsf{U}\times \mathsf{T}$. We define a new curve on a surface by
using the Darboux frame. This new curve whose vector field $\mathsf{V}$ makes a
constant angle with a fixed direction is called as relatively normal-slant
helix. We give some characterizations for such curves and obtain their axis.
Besides we give some relations between some special curves (general helices,
integral curves, etc.) and relatively normal-slant helices. Moreover, when a
regular surface is given by its implicit or parametric equation, we introduce
the method for generating the relatively normal-slant helix with the chosen
direction and constant angle on the given surface.
