Up until the time your home is scheduled for auction, most lenders would rather work out a compromise that would allow you to get back on track with your mortgage than take your home in foreclosure.

After your lender files a NOD but before they schedule an auction, if you get an offer from a buyer, your lender must consider it.

Bankruptcy stops the foreclosure dead in its tracks.  Once you file a bankruptcy petition, federal law prohibits any debt collectors including your mortgage lender, from continuing collection activities.

A deed in lieu of foreclosure is exactly what it sounds like.  The home owner foreclosure signs the deed to the home owner back over to the bank - voluntairly. 

Most loans these days are no longer assumable.  The average mortgage now contains a "due on sale" clause by which the borrower agrees to pay the loan entirely if and when they transfer the property.