ABC News

Former Vice President Dick Cheney is recovering after undergoing heart transplant surgery at Falls Church, Va., today, according to a statement released by Cheney's office.

Cheney has been on the cardiac transplant list for more than 20 months.

"Although the former Vice President and his family do not know the identity of the donor, they will be forever grateful for this lifesaving gift. Former Vice President Cheney is thankful to the teams of doctors and other medical professionals at Inova Fairfax and George Washington University Hospital for their continued outstanding care," the statement said.

Mitt Romney tweeted, "Ann and I send our thoughts and prayers to Vice President Cheney for a fast and full recovery."

Cheney has had five heart attacks, the first of which was in 1978 when he was 37 years old, and the fourth in November 2000, after he and former President George W. Bush were elected to the White House.

In 2001, Cheney had a pacemaker installed into his chest, and in September 2009, he underwent elective back surgery to treat lumbar spinal stenosis.

Cheney's fifth and most recent heart attack was in February 2010.The heart attack and subsequent two-day hospital stay forced the former vice president to miss a breakfast with his former boss, George W. Bush, and hundreds of former White House and campaign staffers. The event, sponsored by the Bush-Cheney Alumni Association, was to be the first time the two met in person since they left office in January 2009.

Instead, Bush dropped by to visit Cheney at his home in Virginia on Feb. 25. Cheney and Bush, both sporting dark suits, shook hands and exchanged grins on the steps of Cheney's residence in McLean, Va., before turning to wave to the ABC News camera.

"Mr. President, welcome," began Cheney.

"Lookin' good," said Bush again.

"Could be worse," the often dry-witted Cheney said.

Cheney underwent surgery in July 2010 to have a heart pump, called a Left Ventricular Assist Device [LVAD], implanted to treat his recurring heart disease.

The LVAD is implanted next to the heart to help its main pumping chamber, the left ventricle, pump blood through the body. Such devices are used mainly for short periods, to buy time for potential transplant candidates as they await a donor organ.

Cardiologists said then that in Cheney's case, the pump was likely a "bridge" that would keep him alive until he could receive a heart transplant. Many cardiac experts said at the time of his surgery that Cheney may be only one step away from a transplant but could find himself on a wait list for "months or years."

Cheney served as Secretary of Defense under George H.W. Bush.

ABC News' Karen Travers contributed to this report.