I have created a path, then filled it. The background (area between the filled path and the rectangular bounding box) is white. How do I set it to be transparent? I tried setting the clipping path, but it didn't work. The attached image has two layers with the same object.

It's already transparent, if you export it to png you will see black and transparent.

…and remember to use 'File > Export Bitmap…' to preserve the default transparent background of the SVG file
[ i.e. do not use 'File > Save as…' -> Cairo PNG (*.png) ]

Thanks for your replies! I finally figured out what the problem was. Namely, once I drew the shape and filled it, I just Copy-Pasted it (Control-C, Control-V). For some reason, it included the white background (which is supposedly actually transparent). In order to avoid that, I discovered that you have to use the "Duplicate" option instead. Works great now!

You are working on Mac OS X? ;)

It is a known issue that Copy&Paste in Inkscape running on OS X (Snow) Leopard under current X11/Xquartz versions causes the pasted data to be inserted as bitmap copy instead of vector data. You need to change your X11 preferences for the pasteboard:

Inkscape FAQ " Copying and pasting in Inkscape creates pixellated images instead of copying the vector objects"
Screenshot: X11 pasteboard preferences

Bug reports:
Bug #307005 in Inkscape: “[OS X] copy & paste inserts a bitmap copy instead of the vector object”
XQuartz ticket #403: "Copy/Paste in Inkscape pastes bitmaps"