You can create 3D curves based on the surface of a model, or create 2D curves on the screen plane. See also Curves.
To extend a curve, connect curves together, close a curve in a loop, change the shape of a curve, or smooth bumps in the curve, see Modify curves.
Once you have curves drawn in the scene, you can use them as a guide for your sculpting or painting. See Sculpt or paint using curves.
To create a 3D curve on a mesh
A Curve object is added to the Object List for each separate curve that you draw.
As you tumble the camera and adjust your view, notice that the curves stay attached to the surface of your model, letting you wrap lines around the object.
To create a 2D curve on the view plane
You can also draw 2D curves without turning on Draw on Plane if you start drawing the curve in an empty area of the 3D View (not touching a model).
Mudbox creates the curve (or your drawn curve displays) on the view plane, and a ViewCurve object is added to the Object List. You can later adjust the scene view to position your model under the 2D curves and apply sculpting or painting based on the 2D curves.
Hiding curves can be useful, for example, if you're using Sculpt on Curve and want to inspect the sculpt or paint strokes on your model more closely.