The Roman Catholic Church has given the name "the incorruptibles" to the bodies of about 100 saints whose bodies are seemingly impervious to decay. Included in this list are St. Bernadette of Lourdes, St. Francis Xavier, and St. Clare of Assissi.
About half of these bodies reside in mausoleums in Italy, and the rest scattered around the world (though mostly in Europe). Many incorruptible bodies remain intact until this day. Some are amazingly preserved, the skin still supple, some have darkened skin, some have shriveled. Some however, have shown signs of decay, and others have been re-buried after the normal process of decay is underway.
The Vatican has recently called on a group of scientists and doctors to examine more carefully the bodies of these saints to give a more scientific view of why these bodies are not in the process of corruption. The first thing comes to mind, of course, is the possibility that the bodies were mummified.
Indeed, in some cases, the bodies have been purposefully mummified, or extensively embalmed. In the case of St. Margaret of Cortona who died in 1297, the preservation of her body was actually requested by the townspeople, and is a matter of public record. However, this fact was forgotten in the intervening time, and people simply assumed that the incorruptibility of her body was an act of God.
In this light, the incorruptibility of the body is not accepted anymore as one of the miracles the Vatican considers as a prerequisite to sainthood.
In other cases, however, the bodies were clearly not preserved by human hands, and the cause of the bodies' preservation is not clear. Some people posit that the unusually cool temperature under church floors where many of the saints were buried contributed to the preservation of the bodies.
But to people of a simpler faith, this phenomenon is an extraordinary gift that God bestowed on his faithful servants. A proof that it is sin that corrupts, not death.
A website with pictures, for the morbidly curious: