The International Olympic Committee (IOC) will delegate power to individual sports' governing bodies to decide if Russian athletes are clean and should be allowed to take part in the respective 2016 Olympics events.
The decision follows a report commissioned by Richard McLaren which said Russia operated a state-sponsored doping programme from 2011 to 2015.
Russian athletes who want to take part in the events will have to meet strict criteria set by the IOC. Track and field athletes have already been banned.
"We have set the bar to the limit by establishing a number of very strict criteria which every Russian athlete will have to fulfill if he or she wants to participate in the Olympic Games Rio 2016.
"I think in this way, we have balanced on the one hand, the desire and need for collective responsibility versus the right to individual justice of every individual athlete.", IOC president Thomas Bach said.
The United States Anti-Doping Agency (Usada) chided the IOC for "refusing to take decisive leadership" whill UK Sports Minister Tracey Crouch voiced out the "need for stronger sanctions".
Russian Sports Minister Vitaly Mutko described the decision as "objective" but "very tough"
The 28 individual sports governing bodies will now have just 12 days to carry out an individual analysis of each athlete.
The International Tennis Federation has confirmed on Sunday that Russia's seven nominated tennis players meet the IOC requirements.
The International Association of Athletics Federations (IAAF) has already ruled that Russian track and field competitors will not compete in the event.
Russia's full Olympic team would consist of 387 athletes.
Whistle-blower banned from Rio
Yulia Stepanova, the Russian whistleblower who exposed the doping crisis will not be allowed to compete in Rio as a neutral athlete, said IOC in a statement. She will instead be invited to Rio as a 'guest'.
Stepanova has previously failed a doping test.
The IOC statement added: "The executive board would like to express its appreciation for Mrs Stepanova's contribution to the fight against doping and to the integrity of sport."
Usada chief Travis Tygart called the decision "incomprehensible", adding it will "undoubtedly deter whistleblowers in the future from coming forward".