In a statement, CAS stated that in the 28 cases, "the evidence collected was found to be insufficient to establish that an anti-doping rule violation (ADRV) was committed by the athletes concerned".

The athletes involved may now have the chance to compete at Pyeongchang 2018.

The International Olympic Committee (IOC) had deemed that "only clean Russian athletes" would be able to compete.

A further 11 cases were found to have enough evidence to uphold the sanctions.

In these cases, the IOC Disciplinary Commission's decisions have been confirmed, but the athletes will not be handed a life ban from the Olympic Games.

Instead, they will only be banned from Pyeongchang 2018.

"The procedures were conducted jointly on an expedited basis and a combined hearing took place from 22 to 27 January 2018 in Geneva," a CAS statement read.

"Every athlete attended the hearing, except two who were not available, and were heard individually.

"Several experts and fact witnesses, such as Dr Grigory Rodchenkov and Prof. Richard McLaren, testified during the hearing.

"Both CAS panels unanimously found that the evidence put forward by the IOC in relation to this matter did not have the same weight in each individual case.

"In 28 cases, the evidence collected was found to be insufficient to establish that an anti-doping rule violation (ADRV) was committed by the athletes concerned.

"With respect to these 28 athletes, the appeals are upheld, the sanctions annulled and their individual results achieved in Sochi 2014 are reinstated.

"In 11 cases, the evidence collected was found to be sufficient to establish an individual ADRV.

"The IOC decisions in these matters are confirmed, with one exception: the athletes are declared ineligible for the next edition of the Olympic Winter Games (i.e. Pyeongchang 2018) instead of a life ban from all Olympic Games.

"The mandate of the CAS Panels was not to determine generally whether there was an organised scheme allowing the manipulation of doping control samples in the Sochi laboratory but was strictly limited to dealing with 39 individual cases and to assess the evidence applicable to each athlete on an individual basis."

The verdict will go down as a major defeat for the IOC at CAS.

The decision is likely to dissuade other International Federations from pursuing sanctions. 

Bobsleigh athletes Dmitry Trunenkov, Aleksei Negodailo, Olga Stulneva and Liudmila Udobkina have had their sanctions overturned, along with skeleton's Aleksander Tretiakov, Sergei Chudinov, Elena Nikitina, Olga Potylitsyna and Maria Orlova.

Alexander Legkov, Evgeniy Belov, Maxim Vylegzhanin, Alexey Petukhov, Nikita Kryukov, Alexander Bessmertnykh, Evgenia Shapovalova and Natalia Matveeva are the cross-country skiers to have been successful in their appeals.

Speed skaters Olga Fatkulina, Alexander Rumyantsev, Ivan Skobrev and Artem Kuznetcov, along with lugers Tatyana Ivanova and Albert Demchenko, have also had sanctions overturned.

Ice hockey players Ekaterina Lebedeva, Ekaterina Pashkevich, Tatiana Burina, Anna Shchukina and Ekaterina Smolentseva are the final athletes to succeed in winning their appeals.

Anti-doping rule violations were confirmed against bobsleigh athletes Aleksandr Zubkov, Alexey Voevoda, Alexander Kasyanov, Aleksei Pushkarev and Ilvir Khuzin.

Cross-country skiers Julia Ivanova, Yulia Chekaleva and Anastasia Dotsenko have also had violations confirmed, as well as ice hockey players Galina Skiba, Anna Shibanova and Inna Dyubanok.

Several of the athletes to overturn their sanctions earned medals at Sochi 2014, including men's skeleton gold medallist Tretiakov.

Nikitina won bronze in the women's event.

Cross-country skier Legkov earned a gold and silver at the Games, while team-mate Vylegzhanin secured three silvers.

Kryukov and Bessmertnykh both earned one silver medal.

Speed skater Fatkulina and luger Ivanova are also set to have their silver medals reinstated, while Demchenko is due to keep the men's singles and mixed team luge silver.

Two members of Russia's gold medal winning four-man bobsleigh crew, Trunenkov and Negodailo were both successful in their appeals.

However, their pilot Zubkov and Kasyanov have had their sanctions confirmed.

Rodchenkov "testified fully and credibly" at the hearings according to his lawyer Jim Walden, who added the verdict "emboldens cheaters". 

"Dr. Rodchenkov testified fully and credibly at CAS," Walden wrote in a statement.

"His truth has been verified by forensic evidence, other whistleblowers, and, more recently, recovery of the Moscow lab's secret database, showing thousands of dirty tests that were covered up.

"This panel's unfortunate decision provides a very small measure of punishment for some athletes but a complete 'get out of jail free card' for most.

"Thus, the CAS decision only emboldens cheaters, makes it harder for clean athletes to win, and provides yet another ill-gotten gain for the corrupt Russian doping system generally, and Putin specifically."

Russian Sports Minister Pavel Kolobkov has called on the IOC to allow the Russian athletes who have overturned their sanctions the chance to compete at Pyeongchang 2018.

"I expect that the International Olympic Committee will accept the CAS decision and grant them the unconditional right to take part in the upcoming Olympics," he told the Russian state news agency TASS.

"Over the past year they have gone through hard times.

"Now they want to move forward and continue doing what they enjoy most of all - to compete in a fair fight."

Russian Olympic Committee President Alexander Zhukov has reportedly described the CAS decision as "fair". 

"From the very start, we've insisted that our athletes are not involved in any doping frauds, and now we are happy that the court has restored their name and all rewards were returned to them," Zhukov said, according to RT.

Aleksandra Brilliantova, head of the ROC’s legal department, added further legal proceedings could potentially be taken by the athletes.

"We were counting on a fair CAS decision and are very pleased that it turned out to be so,” she said, according to the ROC website.

“It is important to understand that the decision of the CAS does not oblige the International Olympic Committee to issue invitations to the Games in Pyeongchang to justifiable athletes.

“In the near future we will hold a meeting with lawyers and representatives of athletes to determine further legal steps in connection with the CAS decision.

“These can be appeals to the CAS ad hoc division, as well as to other appellate instances.

“But the first thing we have to do now is to send an appeal to the IOC with a request to consider the issue of inviting those athletes who have passed the Olympic selection to the Games in Pyeongchang and in respect of which there is a positive CAS decision.”

Cases involving biathletes Olga Zaytseva, Olga Vilukhina and Yana Romanova are still to take place, having been suspended until after Pyeongchang 2018.

Bobsleigh's Maxim Belugin is the one Russian athlete not to have appealed following his disqualification.

He is believed to have produced a positive test result following reanalysis of his samples, although full details of his case have not been disclosed.

A table of athletes sanctioned by the IOC Disciplinary Commission, their Sochi 2014 results and the outcome of today's CAS verdicts