Spain has recorded its lowest death toll in two months with 87 deaths in 24 hours, while Italy recorded its lowest daily toll, 145, since lockdown was declared.
It was the first time that Spain announced under 100 deaths in a day since mid-March, which could be due to a delay in reporting over the weekend, the health emergency coordinator said.
"This is good news – even if it is always hard to speak about deaths from this disease," said Fernando Simon, Spain's head of health emergency coordination.
There have been a total of 231,350 cases of coronavirus in Spain and 27,650 deaths. There were just over 400 newly confirmed cases of COVID-19 reported on Sunday.
Prime Minister Pedro Sanchez said on Saturday that the government would request to extend the state of emergency for another month as the country continues to phase out restrictions on movement amid the coronavirus pandemic.
The state of emergency was set to end on May 24.
Declaring the lowest number of daily deaths since 9 March, Italy is now looking to bring forward the reopening of some commercial activity in the country, with retailers, hairdressers, salons, restaurants and cake shops authorised to open from tomorrow.
31,908 people have died and 224,760 have been infected in Italy.
Latest coronavirus news:
Boris Johnson says he understands people are 'frustrated' by social distancing rules
South Korea confirmed just 13 new cases
Italy will allow travel between regions and abroad from June 3
The UK records another 468 deaths, bringing the total to 34,466
Boris Johnson acknowledges 'challenge' of lockdown
Boris Johnson implored the British public to continue social distancing while acknowledging the "challenge" of lockdown.
Writing in the Mail on Sunday, Johnson said the public's "sacrifices have paid off" but that people would have to continue social distancing.
The UK prime minister said he did not "underestimate how difficult it has been for everyone to be cut off from friends and parents, children and grandchildren, brothers and sisters...those vital human connections have been cruelly denied to all of us by this insidious disease."
But the UK has not yet met all five tests that allow them to begin phasing out more distancing measures, he added.
People in England are now able to spend more time outdoors and begin returning to work but schools and most businesses remain closed.
Health authorities on Sunday reported 170 new deaths, bringing the total to 34,636 - the second highest number in the world.
At the daily government briefing, Business Secretary Alok Sharma said the first clinical trials of the vaccine being conducted by Oxford University are progressing "well".
Arrests as protests held in some European countries
Police have been forced to intervene at a number of demonstrations in several cities across Europe, as groups of people gathered to protest restrictions put in place to combat the spread of coronavirus, which has now killed more than 300,000 people.
Tear gas was used on protesters in Poland and arrests were made at a gathering in London's Hyde Park. Authorities had to police a number of gatherings in German cities, attempting to ensure social distancing measures were observed.
Protestors are upset at restrictions which have harmed economies, and are perceived to be limiting freedoms. In some cases, conspiracy theories and anti-vaccine views were aired.
Italy set to ease travel restrictions
Italy will ease travel restrictions from June 3 allowing travel between regions and some international travel.
People will be able to travel within their own region from Monday after the country gradually began lifting lockdown restrictions in early May.
New travel restrictions will only be put in place depending on certain areas and their outbreaks of the coronavirus, Italy's Council of Ministers said.
Italy was the first country outside of Asia to have a serious outbreak of COVID-19 and issued lockdown restrictions in early March.
Starting in May, the country gradually let many people return to work as long as they respect physical distancing measures.
Italy has the third highest death toll in the world after the United States and United Kingdom with more than 31,000 people who have died.
South Korea records just 13 new COVID-19 cases
South Korea recorded just 13 new cases of coronavirus, giving hope that a cluster of new cases linked to nightclubs has been managed.
There have been a total of 11,050 cases in South Korea and 262 deaths, according to the Korea Centres for Disease Control and Prevention.
The agency did not say if the 13 new cases were linked to the outbreak in nightclubs which were linked to more than 150 confirmed cases.
Record death toll in Russia
Meanwhile, Russia recorded a record number of deaths on Saturday, with 119 in 24 hours, AFP reported.
The country had around 9,000 new confirmed cases of coronavirus bringing the total number to more than 272,000 cases.
Russia has the second-most cases in the world after the United States but a much lower death toll, with the official number at just over 2,500 deaths.
Some have called into doubt this number, an accusation that authorities have rejected.
Greece set to open public beaches
Greece is set to reopen public beaches amid a heat wave.
The government put in place strict social distancing measures that include a steep fine and potential closure for businesses that violate the rules.
Group sports and food operations are still prohibited in Greece but the government is considering opening bars and restaurants in late May.
This article is being updated as new coronavirus updates emerge.