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Like lead based paint, many older homes have asbestos.  Asbestos was used as an insulator and mostly found wrapped around duct-work, plumbing lines and other pipes, floor tile, attics, roofing, etc.  If exposed and inhaled over a long period of time, it can severely damage human lungs. Asbestos can also cause other problems such as warts, plaques, and asbestosis.  Although it has been banned in many other countries, the Unites States still allows its use in new construction.

Removal 

Asbestos should be handled and removed by professionals.  It can be encapsulated so that the tiny fibers will not escape into the air.  It should not be vacuumed as the fibers are too small for the vacuum.   The EPA website is a good resource on what to do when you think you have exposed this material.  There you can find a local asbestos mitigation service.