It is often stated in the paleontological literature that the chance an animal will become fossilized is "one in a million." This number is meant to be taken figuratively, the point being that the odds of surviving the rigors of deep time are extremely remote. Nevertheless, all field paleontologists know that the earth is biased when it comes to giving up its dead--the odds of an animal being preserved and consequently exhumed are much greater in some settings than others.
Studies by taphonomists (paleontologists who study the transition of animals from the biosphere to the lithosphere; taphonomy literally means "burial laws") have shown that organisms that die on land in lush jungle locales are rarely fossilized. In these settings, there is little chance of being buried, scavenging vertebrates and insects are prevalent, bacteria that break down flesh and bones are abundant, and the soils are extremely acidic and tend to dissolve bones. As a result, remains of dinosaurs from such former surroundings are practically nonexistent. Conversely, dinosaurs are commonly found in areas that were once fluvial settings and in regions of extreme aridity. In the former case, it is clear that dinosaur remains were rapidly buried before substantial scavenging could take place. Remains of dinosaurs that were washed into the fluvial systems are found buried in actual river channels, whereas others are found out on the former floodplains at the location where they fell and were covered by sediments from floodwaters that breached river banks. Because river currents tend to scatter and break up bones, remains from river channels are often biased toward certain bones depending on the strength of the current. (Such aggregations are called Voorhies groups after one of the first paleontologists to study the phenomenon by which certain bones, such as ribs and vertebrae, tend to readily tumble downstream, leaving behind only partial skeletons.) Dinosaur fossils found on former floodplains also often show bias toward elements such as pelvises and larger long bones that were difficult for scavenging or predaceous theropod dinosaurs to consume.
In any event, once bones were entombed in fluvial sediments, not only were they protected from scavengers and many types of bioorganisms, but they could also begin a process known as permineralization. Water percolating through the sands or muds was often rich in silica (natural glass) and other minerals, which could infill the pores of the bones and make them physically resistant to crushing by the overlying sediment. At least some minor replacement of the actual bone matrix usually occurred as well, typically by iron-rich minerals, but it should be noted that most dinosaur bones actually retain much of the original calcium and phosphatic minerals they possessed in life. As such, the phrase "turned to stone"--often used to describe fossil bone--is misleading.
Dinosaurs dying in arid regions also stood a reasonable chance of becoming fossilized. Aridity tends to desiccate a carcass, making it less attractive to scavengers. And unlike jungle or forest settings, deserts have considerably fewer organisms suited for the breakdown of animal tissues. Windblown sands, as well as drifting and collapsing sand dunes, were agents of burial for such animals. Subsequent rainfall during the wet seasons carried minerals into the buried bones.
If dinosaur remains entombed in the ways described above did not later become metamorphosed (modified by upheavals of the earth) there is a good chance they are still around today, thus enabling the details of their burial to be pondered by taphonomists, either professional or amateur.
This answer was originally posted on September 16, 2002.