How can I add thickness to a curve? I already tried with the extrude and depth options in the curve settings, but it doesn't work properly.
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All you need to do is set the Fill to Full on the curve to get a complete 3D shape. It defaults to Half, which is why you only have half of you shape showing.
Add a nurbs circle and use it as bevel object:
You can control the thickness by scaling the circle.
The simplest way to do this is to use a Solidify modifier.
This is the result from the default Bezier curve.