For some serial killers, murder isn’t enough. They tease, boast, taunt, challenge, mystify and torment police, too. Sometimes, they even beg to be stopped.

The serial sniper around Washington, D.C., is the latest in a line of serial killers who feel compelled to communicate with their pursuers.

Jack the Ripper bragged to British police about his murders in a series of letters. David Berkowitz declared memorably in a note, “I am a monster. I am the Son of Sam.” William Heirens, Chicago’s Lipstick Killer, scrawled on a victim’s wall, “For heaven’s sake, catch me before I kill more.”

For the killers, such messages often satisfy their morbid and desperate egos, heighten their sense of command over the world, or intensify the rush of their cat-and-mouse game with authorities, crime experts say. For the police, the messages may carry a wealth of clues, from DNA on a licked envelope to poor English that can help build the killer’s profile.

The messages bring difficult decisions about whether to release their content to protect the public or help find the killer, at the risk of stirring his anger or promoting copycats.

But sometimes, these communiques may help solve the case.

Like some of his serial predecessors, the Washington sniper has revealed himself to police in a megalomaniacal voice. He has left a tarot card representing death and saying, “Dear policeman, I am God.” A message over the weekend warned: “Your children are not safe anywhere at any time.”

Reacting to a phone call, police have tried to engage the unknown sniper through the media, asking him in an extraordinary plea to call back and clarify a garbled message.

Bill Clark, a former New York City police detective who is now an executive producer of “NYPD Blue,” said serial killers who leave messages “absolutely want the recognition for what they’re doing and want attention.”

“They want to show they’re in control,” he said.