How to vary the thickness of a Bezier curve? I have made a bezier tube and I want some parts of it to be large and others to be small.

When I try to set the depth for one of its vertices, it just changes the depth for the whole object. Is there a way to change the depth for separate parts of the bezier curve?
Vertex Radius
You can scale the thickness of a curve at the selected vertices with AltS in edit mode.
Taper Objects
You can also use a Taper object:

The distance from the object origin (yellow dot) in the local Y direction controls the width of the main curve: