The Court of Arbitration for Sport (CAS) on January 22 began proceedings that are expected to run for six days at a conference center in Geneva.
Because of the size of the hearings, the CAS was forced to temporarily move from its small headquarters in the Swiss city of Lausanne to the larger Geneva conference center.
One panel of three judges will hear 28 cases, and a second panel will judge 11. Two of the judges -- Christoph Vedder and Dirk-Reiner Martens, both from Germany -- will sit on both three-person panels, the court said.
Two witnesses who played key roles in exposing doping practices that led to the bans, whistle-blower Grigory Rodchenkov and World Anti-Doping Agency (WADA) investigator Richard McLaren, were expected to testify by video or teleconference at the closed-door hearings.
Verdicts are expected by February 2, a week before the Pyeongchang Winter Olympics are due to begin in South Korea.
The IOC has banned Russia from the Games, citing its 'unprecedented systematic manipulation' of the Olympic anti-doping system, but some Russians are likely to be allowed to compete as neutral athletes.
Three Russian biathletes have also appealed against the IOC ban and their cases will be heard later.
The IOC imposed lifetime bans on 43 Russians in all for doping. Maksim Belugin, a member of bobsleigh teams that finished in fourth place at Sochi in 2014, is the only banned athlete not to have lodged an appeal.
The 42 athletes who appealed deny being part of a state-backed doping program that Olympic authorities said was in place during the Sochi Olympics.
All of the athletes were retroactively disqualified from the Sochi games over the doping allegations.
With reporting by AFP, Reuters, The Telegraph, and CNN