Just as government officials had feared, the drug violence raging in Mexico is spilling over into the United States.

U.S. authorities are reporting a spike in killings, kidnappings and home invasions connected to Mexico’s murderous cartels. And to some policymakers’ surprise, much of the violence is happening not in towns along the border, where it was assumed the bloodshed would spread, but a considerable distance away, in places such as Phoenix and Atlanta.

Investigators fear the violence could erupt elsewhere around the country because the Mexican cartels are believed to have set up drug-dealing operations all over the U.S., in such far-flung places as Anchorage, Alaska; Boston; and Sioux Falls, S.D.

“The violence follows the drugs,” said David Cuthbertson, agent in charge of the FBI’s office in the border city of El Paso, Texas.

The violence takes many forms: Drug customers who owe money are kidnapped until they pay up. Cartel employees who don’t deliver the goods or turn over the profits are disciplined through beatings, kidnappings or worse. And drug smugglers kidnap illegal immigrants in clashes with human smugglers over the use of secret routes from Mexico.

So far, the violence is nowhere near as grisly as the mayhem in Mexico, which has witnessed beheadings, assassinations of police officers and soldiers, and mass killings in which the bodies were arranged to send a message. But law enforcement officials worry the violence on this side could escalate.

“They are capable of doing about anything,” said Rusty Payne, a Drug Enforcement Administration spokesman in Washington. “When you are willing to chop heads off, put them in an ice chest and drop them off at a police precinct, or roll a head into a disco, put beheadings on YouTube as a warning,” very little is off limits.

In an apartment near Birmingham, Ala., police found five men with their throats slit in August. They had apparently been tortured with electric shocks before being killed in a murder-for-hire orchestrated by a Mexican drug organization over a drug debt of about $400,000.

In Phoenix, 150 miles north of the Mexican border, police have reported a sharp increase in kidnappings and home invasions, with about 350 each year for the last two years, and say the majority were committed at the behest of the Mexican drug gangs.

In June, heavily armed men stormed a Phoenix house and fired randomly, killing one person. Police believe it was the work of Mexican drug organizations.

Authorities in Atlanta are also seeing an increase in drug-related kidnappings tied to Mexican cartels. Estimates of how many such crimes are being committed are hard to come by because many victims are connected to the cartels and unwilling to go to the police, said Rodney G. Benson, DEA agent in charge in Atlanta.

Agents said they have rarely seen such brutality in the U.S. since the “Miami Vice” years of the 1980s, when Colombian cartels had the corner on the cocaine market in Florida.

Last summer, Atlanta-area police found a Dominican man who had been beaten, bound, gagged and chained to a wall in a quiet, middle-class neighborhood in Lilburn, Ga. The 31-year-old Rhode Island resident owed $300,000 to Mexico’s Gulf Cartel, Benson said.