The good news is that menstruation should have limited impact on your performance. In fact, women have run well and even set records during all phases of the menstrual cycle.
Don't be afraid to run when you're menstruating. You may find that running can actually improve your mood and alleviate physical symptoms before and during your period. If you're training for a big race, it's recommended that you plan to run when you have your period so you can see how it feels. It will alleviate some of the fear and nervousness if you have it for your race. Tampons are recommended over pads for comfort and to avoid chafing. Just make sure you carry an extra one on your long runs during that time of the month.
Some female athletes may develop menstrual irregularities, especially if they have very low body fat, heavy training, and poor nutrition. They may stop menstruating -- a condition known as athletic amenorrhea, which can lead to other issues such as infertility and stress fractures. You should discuss any menstrual irregularities with your health care provider.
ACE Personal Trainer Manual: The Ultimate Resource for Fitness Professionals, 3rd edition. American Council on Exercise. 2003
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Hyperhidrosis. U.S National Library of Medicine and the NIH.
Hosey, Robert, M.D., et. al., "Exercise-Induced Anaphylaxis and Urticaria" American Family Physician, October 15, 2001
National Institute of Diabetes and Digestive and Kidney Diseases, NIH. "What I need to Know About Bladder Control for Women"