This article is from the Scuba Diving FAQ, by njs@scifi.maid.com (Nick Simicich) with numerous contributions by others.
Scuba diving is a physically demanding sport, which requires a healthy
heart, well able to tolerate exercise, and healthy lungs.
Additionally, any illness which might incapacitate you, such as with a
siezure, or with unconsciousness, such as uncontrolled fainting. There
are many medical conditions which are considered disqualifying for
scuba diving. The Diver's Alert Network (phone +1.919.684.2948) will
provide over-the-phone advice about medicine, medications, diving, and
their interaction, as well as assisting you in finding the appropriate
chamber or a local doctor who is familiar with diving medicine and so
forth, and is a worthwhile organization to join.
Some medical conditions which are generally considered disqualifying
(although there are exceptions for well controlled conditions, in some
cases, consult your doctor) are asthma, diabetes, epilepsy or any
other siezure disorder, history of spontaneous (or, from some sources,
any) pneumothorax, emphesema, heart illness which inhibits your
ability to exercise to a certain level, and others.
There is some experimental evidence that diving while pregnant could
be dangerous for the fetus, so it is contraindicated. This is a
compressed air issue, so shallow, reasonable snorkeling should be
fine, if your doctor says you can tolerate exercise and swimming.
 
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