1. To surround a building, a city, etc. with soldiers and to try to take control of it: The military besieged the castle for several days before taking possession of it.
2. To gather around someone is a way that is disturbing, annoying, or aggressive, etc.: The political candidate was besieged by reporters of newspapers and TV when he announced that he would not be running for office.
3. To overwhelm a person with too many questions or requests for something: The sales department was besieged with hundreds of customers who wanted to know why the store ran out of computers so quickly on the first day of the special reduction in prices.