In this paper, we consider the Darboux frame of a curve
α
lying on an arbitrary regular surface and we use its unit osculator Darboux vector
D
¯
o
, unit rectifying Darboux vector
D
¯
r
, and unit normal Darboux vector
D
¯
n
to define some direction curves such as
D
¯
o
-direction curve,
D
¯
r
-direction curve, and
D
¯
n
-direction curve, respectively. We prove some relationships between
α
and these associated curves. Especially, the necessary and sufficient conditions for each direction curve to be a general helix, a spherical curve, and a curve with constant torsion are found. In addition to this, we have seen the cases where the Darboux invariants
δ
o
,
δ
r
, and
δ
n
are, respectively, zero. Finally, we enrich our study by giving some examples.