The Russians are seeking help from the West as evidence grows that they are facing a common enemy: the austere Wahhabi Muslim sect, which has links to armed uprisings from Algeria to Afghanistan.
From their base at Medina University in Saudi Arabia, according to Salih Brandt, a spokesman for the Chechen government abroad, the Wahhabis are now spreading their influence to the Caucasus, where Russian forces are trying to drive out Islamic fighters who have entered Dagestan from neighbouring Chechnya.
Panic in Russia was intensified by the initial absence of any claims of responsibility for the bombings. Suspicion immediately fell on Islamic rebels in the Caucasus, but evidence was hard to find. The Chechen government denied involvement.
After bombs destroyed two blocks of flats in Buinaksk, Dagestan, and in the Moscow suburb of Pechatniki, killing 64 and 92 people respectively, a man with a Caucasian accent phoned Interfax news agency saying: "What happened in Buinaksk and Moscow is our answer to what the Russian army is doing in Dagestan."
The Russian Interior Minister, Vladimir Rushailo, said the tape would be studied carefully and calmly; the authorities were not jumping to conclusions.
After another bomb flattened a block on Moscow's Kashirskoye Shosse last Monday, killing 118, there was a more specific call to Itar-Tass news agency. A man claiming to represent the little-known Dagestan Liberation Army (DLA) said the bombings were to avenge the deaths of Muslim women and children in Russian air raids over Dagestan. "We will answer death with death," he said.
The Caucasian connection immediately drew attention to Shamil Basayev, the Chechen warlord leading the fight in Dagestan, who first attained notoriety in Russia by taking civilians hostage in a hospital. He denied involvement in the bombings, saying that hurting civilians was not his style. But his comrade, the fighter Amir al-Khattab, said the attacks were a response to what the Russians had done in Karamakhi and Chabanmakhi, two Dagestani villages where followers of the Wahhabi sect were living until federal forces bombed them out this month.
Mr Brandt said Mr Khattab was involved in the aftermath of the Afghan war, a close associate of millionaire terrorist Osama Bin Laden, and well- connected in the radical Islamist world. "He has moved into Chechnya with large sums of money, fighters - mainly from Afghanistan, Algeria, Tajikistan and Arab countries - training facilities, and the Wahhabis' austere ideology," he said.
"The spread of Wahhabi influence into Chechnya and Dagestan follows a familiar pattern. In mosques, student groups and Islamic centres round the world there are graduates who have been given free tuition at Medina University, expounding the Wahhabi creed.
"The most public expression of its beliefs is in Afghanistan, where the ruling Taliban has sent women home from their jobs, forced men to grow beards and beaten those who fail to attend prayers in the mosques. None of this is prescribed in the Koran or in the practice of the Prophet Mohammed."
If there is a pattern to the bombs in Russia, it is that they come after the militants have suffered defeats in Dagestan. But Russia's Prime Minister, Vladimir Putin, has also spoken of the involvement of "international saboteurs". A pointer to the possible involvement of Osama Bin Laden, say experts, is that the block bombed in the southern city of Volgodonsk on Thursday suffered the same kind of damage as the US barracks destroyed in Dhahran, Saudi Arabia, in 1996, a bombing to which the Saudi mastermind is linked.
The modus operandi of the bombers in Moscow has been distinctive. Experts believe that at least one bomber has experience in the construction and demolition industry, as the blocks of flats collapsed in the same way.
The bombers, investigators say, hired commercial storage space in the buildings they planned to blow up and, over time, brought in sacks of explosives. A cache of these sacks was discovered, but more might still be in Moscow. Newspaper advertisements sought van drivers to deliver "sacks of sugar" to different parts of the city.
Although Mr Khattab is regarded as something of an outsider in the Caucasus, and therefore might lack the resources to carry out bombings far away in the Russian federation, many fear he has increasing influence over Mr Basayev, even though the two are very different.
Shamil Basayev is motivated by hatred of Russia for what it did in the 1994-1996 war against Chechnya, and for its refusal to consider Chechen independence. He dreams of restoring the Islamic state that existed in the Caucasus before the tsars conquered the region in the 19th century. Mr Khattab is a mysterious figure who came originally from either Saudi Arabia or Jordan. He is married to a Dagestani Wahhabi, and it has been reported that his father-in-law was killed in the siege of Karamakhi and Chabanmahki. He could therefore have a personal desire for revenge on the Russians.
The Wahhabis first moved into the villages a year or two ago, rejecting the jurisdiction both of the Russians and the local Islamic authorities: the Mufti of Dagestan even declared a fatwa against them. When Chechen militants first invaded the craggy Botlikh district of Dagestan in early August, the Wahhabis were accused of joining them and declaring a holy war for Dagestani independence.
After the Chechens were expelled at the end of August, the Russians decided it was time to disarm the Wahhabis, but this turned out to be a far more difficult operation than they had expected. Karamakhi and Chabanmakhi had been turned into strongholds, with networks of bunkers and tunnels, and the Russians had to resort to a heavy aerial bombardment to force the Wahhabis out.
At this point, on 4 September, the apartment block in Buinaksk, home to Russian army officers and their families, was bombed. The Chechens seized their chance for a fresh invasion of Dagestan, this time in the Novolaksk district. By the end of last week the Russians seemed to be getting the upper hand, but they have learnt not to declare victory prematurely.