Generally speaking, any person who is not a United States citizen is subject to an immigration review every time they enter the United States. It is at this re-entry where you can be deemed inadmissible and denied entry, even if you were admitted previously. If you have some sort of immigration benefit pending you may be deemed inadmissible upon re-entry, or your prior pending application may be deemed abandoned upon attempting to re-enter.
Asylum applicants, asylees, refugees, and permanent residents who obtained such status based upon asylum or refugee status, should also ensure they receive permission prior to departing the country. Depending on your circumstances, and current visa status, you may need to file for Advanced Parole prior to leaving the United States, to ensure you are permitted re-entry. It is extremely important to file this document, and receive approval, prior to leaving the United States.
In emergency situations you can seek to have the application expedited. Generally, this is considered to be an emergency situation, severe financial loss to a company or person, death in the family, etc.
At Shupe-Dhawan when we file Adjustment of Status Applications, we generally also request Advanced Parole, as well as Employment Authorization Card simultaneously. This ensures permission to work and permission for re-entry generally within 90 days.