Theories of low-mass star formation make a number of predictions for the structure of the envelopes of gas and dust around embedded young protostars. With the submillimeter array camera SCUBA on the JCMT we have been able to test these theories by imaging the submm dust continuum emission and directly measuring the density profiles in protostellar envelopes. Some of the youngest (Class 0) protostars show structure that deviates significantly from the theoretical predictions, and I will discuss possible explanations of this result.