When should assert() be used?

In developing a large C++ programming project with many developers, we have run into issues with inappropriate use of assert() in the code which results in poor quality where the assertion does indeed occur and the product crashes.

The question is what are good principles to apply to use assert() appropriately? When is it proper to use an assert() and when is it not? Is there a list of criteria that each assertion should pass in order to be legitimate? How can we encourage proper use of assert()?

As a first crack at this, I would say that assert() should only be used to document a condition that is believed to be impossible to reach and which should be identified as an assert() failure at run time where it ever to arise because programming assumptions are being violated.

Can folks do better than this? What is your experience with assert()?