In the news over the past two days, there was a positive test of SARS-CoV2 (COVID-19) in a cat in Belgium. The pet parent was symptomatic prior to the cat developing clinical signs of disease (vomiting, breathing issues and diarrhea). The cat's feces (and vomit) were tested for the novel coronavirus via PCR and came back positive. While it's proven if the cat was ill due to SARS-CoV2, or something unrelated, the cat was not the source of infection for the owner (and both are reportedly doing well). This is one of three reports of potential reverse zoonoses (infection from human to animal).
The risk of humans becoming infected by COVID-19 from an infected pet is extremely low. So far, no research or tests have proven that they can act as a fomite. There is always the possibility that your pet's fur can act as a fomite if an infected individual sneezes or kisses their pet and another family member then touches or kisses the pet. Which is why infected individuals are encouraged to have a healthy individual care for their pet while they are ill. See the CFIA's latest information online here. So far, two dogs have tested positive by PCR who were asymptomatic and living with owners that were positive for SARS-CoV-2. I used Promed mail a lot when I was in school. Here are links to reports for Dog 1, Dog 2, and Cat 1.
For pet parents who wish to stay on top of COVID-19 issues (amongst other infectious organisms) head over the Dr. Scott Weese's blog "Worms and Germs". Dr. Weese is a veterinarian with a specialty in internal medicine and his research interests include emerging infectious diseases.
Edit April 2, 2020:
I posted in my Instagram about the recent report of cats being tested to see if they could spread SARS-CoV-2. According to this study, they can. And ferrets too.