How long before a bankruptcy is discharged?

The answer depends on what chapter you are filing and the circumstances of your case. Typically a Chapter 7 filing will be discharged in 4-6 months. 
A Chapter 13 filing will be discharged in 3-5 years, depending on your income level. There are circumstances, however, that can severely delay discharge and I have seen Chapter 7 cases still not discharged after 6 years because of the actions of the debtor.