Has the FDA approved any Covid-19 vaccines?

The Food and Drug Administration (FDA) has granted Emergency Use Authorization (EUA) to the Pfizer-BioNTech vaccine, the Moderna vaccine, and the Johnson & Johnson vaccine. Additional vaccines are in Phase 3 of clinical trials and maybe submitting data for review with the FDA soon. 

Emergency Use Authorization exists to allow the FDA more flexibility to review and authorize products for use during public health emergencies, like a pandemic.  In order to grant an EUA, the FDA must determine that a vaccine can effectively prevent infection or disease and that its benefits outweigh potential risks (like serious side effects). While the EUA allows people to start receiving the vaccine, vaccine developers will continue to submit data to the FDA to achieve full FDA approval through a standard Biologic License Application (BLA). The BLA approval represents the FDA’s highest standard, but it requires a longer period of evaluation.

Related Questions
What do I do if I have Covid-19 now?
Adhere To CDC Guidelines If you test positive for Covid-19, the following steps -- outlined by the Centers for Disease Control and Prevention (CDC) -- will help you care for yourself and protect others: Alert close contacts. Tell anyone you have recently come into contact with that you have Covid-19, so they can take the necessary precautions and prevent further spread. If you have concerns about confidentiality, especially in the workplace, check with your employer. Many can notify others…
Where should I go for trusted and up-to-date information on Covid-19 vaccines?
The Centers for Disease Control and Prevention (CDC) has great information on the Covid-19 vaccines, including a FAQ page. Another CDC resource will direct you to state Department of Health websites. Those sites will help you determine when you can get the vaccine in your area.
I'm young and healthy. Should I get vaccinated?
While not considered to be at high risk for severe complications from Covid-19, serious cases do occur among younger, healthier people. The Centers for Disease Control and Prevention (CDC) also warns of the potential long-term health effects of Covid-19 infection, even for those who experienced mild illness.