What are the medical requirements for a private pilot?

At a minimum, a private pilot must have a current third-class or higher medical certificate when acting as pilot in command.

A current first-class or second-class medical certificate is an alternative, as these two qualifications are more stringent.  However, a first and second-class medical certificate are generally reserved for commercial operations.  If under 40 years of age, the third-class medical certificate expires after that 60th month.  If 40 years old or older, a third-class medical certificate expires after the 24th month of the date of issue.

Reference(s):

14 CFR 61.23 Medical certificate: Requirements and duration

Other Pilot Certificate Topics