While the US Centers for Disease Control and Prevention (CDC) says there's no evidence that animals play a significant role in spreading the virus, there have been reports of pets being infected.
And since the coronavirus is believed to have infected wild animals before jumping to humans, this is an area scientists need to learn more about in an effort to control the spread of Covid-19 and future illnesses.
Can animals catch coronavirus?
The coronavirus has been identified on at least 25 mink farms in the Netherlands, according to the country's government, which said on Friday that it had culled animals at 24 infected farms.
Covid-19 was found in three of 11 cats at one mink farm, and the government said cats may play a role in the spread of the virus between farms.
At a farm in Teruel, Spain, 92,700 mink are to be culled after 78 of 90 animals tested were found to have the coronavirus -- 87% of the sample.
There have also been reports of pet cats and dogs infected with the virus in several countries, including New York, Hong Kong and the Netherlands. Eight big cats tested positive at New York's Bronx Zoo.
Can I catch coronavirus from my pet?
The CDC says the risk of animals spreading Covid-19 to people is "considered to be low" and the agency does not recommend routine testing of pets.
A YouTube video released by the US Food and Drug Administration (FDA) in mid-June said that "it doesn't seem like animals can give you the virus," although you may be able to give it to them.
However, Dutch authorities said it was "plausible" that a mink may have infected a human with Covid-19, and instituted mandatory testing of animals at all mink farms in the Netherlands.
Since animals can spread other diseases to humans, it's always best to wash your hands after touching a pet and before touching your face.
Didn't the coronavirus originate in wild animals?
Researchers believe the coronavirus spent some time infecting both bats and pangolins before it jumped into humans. Scientists suspect humans first came into contact with an animal sick with the disease at a wet market in China.
But scientists say a third species of animal may have played host to the virus before it spilled over to people.
What is clear is that the coronavirus has swapped genes repeatedly with similar strains infecting bats, pangolins and a possible third species, a team at Duke University, Los Alamos National Laboratory and elsewhere reported in the journal Science Advances in May.
What's also clear is that people need to reduce contact with wild animals that can transmit new infections, they concluded.
The World Health Organization advises anyone visiting live animal or animal product markets to practice hygiene measures including handwashing and avoiding touching your face. Market and slaughterhouse workers should wear protective clothing and regularly disinfect surfaces.
The World Organization for Animal Health does not recommend any Covid-19-related sanitary measures for the international movement of animals or animal products.
How can I protect my pet from Covid-19?
Consider avoiding dog parks and other crowded public places, the FDA advises. The six-foot distancing rule applies to leashed pets, as well as to other people.
The FDA also suggests avoiding contact with animals if you are sick -- if possible, have someone else care for your pet until you're well again, or wear a face covering around them.
If your pet gets sick after contact with a person with Covid-19, call your veterinarian and find out about telemedicine consultations or other plans for seeing pets.
Some pet owners in China have fitted their dogs with tiny face masks, but Hong Kong's Society for the Prevention of Cruelty to Animals said there was no benefit to that, and it was probably fairly distressing.
Animal rights groups warned that a bigger risk than the spread of the virus from animals to humans is the "spread of fear" causing owners to abandon their pets.
Should this change how we behave around animals?
Fur farms such as those where the mink outbreaks occurred are banned in many countries because of concerns around animal welfare and ethics.
The Dutch government is working on a plan in which mink farms can voluntarily close before the planned phase-out date of 2024. Fur Free Alliance, an international coalition of animals rights groups, has called for a global closure.
Research published in Science Advances warned that humans are setting ourselves up to be infected with new viruses by operating "wet markets" where many different species of live animals are caged and sold, and by moving deeper into forests where animals live.
They said "reducing or eliminating direct human contact with wild animals is critical to preventing new coronavirus zoonosis [transmission from animals to humans] in the future."
Jane Goodall, the pioneering chimpanzee expert, said she hoped that the coronavirus would make us reflect on our relationship with the natural world.
She said humans had "disrespected" nature and animals, "and as we destroy the forests and the habitats, species which normally wouldn't interact have been crowded together" and have been forced into closer contact with humans.
Goodall noted that HIV originated with the hunting of chimpanzees, that Middle East respiratory syndrome -- another coronavirus -- comes from camels, and that modern farming practices create ideal conditions for a virus to jump from an animal to a human. The climate crisis could also bring further problems and more diseases.
"So, let's hope we come out of the pandemic and can work out together, a greener future economy, and a better way to live in harmony with the natural world," Goodall said, "for the sake of the environment, animals, our own health and future generations."