In earlier Blender versions, the "Transform Properties" panel showed the object dimensions (i.e., bounding box: width, height, depth). Now, it just shows location, rotation, and scale.
2 Answers
Press N to bring up the Properties Panel of the 3D View, the Dimensions are listed below Scale in the Transform panel (bottom right):
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$\begingroup$ Thanks, I had been looking at the object "Transform" panel instead. $\endgroup$ Nov 4, 2013 at 15:21
While CoDEmanX's answer is still correct for Blender 2.8, you have to have the Modeling view mode (bar at the top) selected and be in object mode for the dimensions to appear in the transform panel. User serdar.sanri added this information here, however this question pops up first in the google search, so it might help to find this comment here, too.
As of Blender 3.0, Modelling view is no longer required.