1institute or conduct legal proceedings against (a person or organization):they were prosecuted for obstructing the highway [no object]:the company didn’t prosecute because of his age
institute legal proceedings in respect of (a claim or offence):the state’s attorney’s office seemed to decide that this was a case worth prosecuting
2continue with (a course of action) with a view to its completion:a serious threat to the government’s ability to prosecute the war
archaic carry on (a trade or pursuit):I am waiting for permission to prosecute my craft