While the Centers for Disease Control (CDC) is marketing widespread use of the vaccines in the U.S. for both old and young alike; many other countries have limited Covid-19 vaccine use under certain conditions. Health officials around the world are giving varying advice on safety issues as Covid-19 vaccines are given to more people, and more information is collected.
The article at the following link provides summaries of some of the concerns that have emerged or been raised by medical officials.
(Source: "(UPDATED) Exclusive Summary: Covid-19 Vaccine Concerns")