Judging by the sheer number of dinosaur eggs that must have been laid during the Triassic, Jurassic and Cretaceous periods, you might think fossilized eggs would vastly outnumber fossilized dinosaurs. Let's do the math: in any given year, hundreds of millions (and probably billions) of dinosaurs roamed the earth's continents, and mature females laid anywhere from two or three to 20 or 30 eggs at a single sitting. Logically, fossilized dinosaur eggs should be as abundant as fresh chicken eggs, not to mention a whole lot more interesting. (See 10 facts about dinosaur eggs and a gallery of dinosaur egg pictures.)
Sadly, though, this isn't the case. The fact is that intact, fossilized dinosaur eggs containing fossilized embryos are phenomenally rare; more often, researchers find scattered pieces of eggshell, from which it's virtually impossible to piece together a complete egg. And even when complete, fossilized dinosaur eggs are found, it's a tricky business to figure out exactly which dinosaur laid them--witness the notorious case of Oviraptor, which for years was accused of stealing the eggs of Protoceratops until further study revealed that it was simply guarding its own nest! (It should be mentioned here that "dinosaur eggs" are regularly offered for sale by unscrupulous fossil dealers, who know full well that they're trafficking in reptile or bird eggs, ordinary rocks, or even the smooth stomach stones of cattle.)
Dinosaur Eggs and Nesting Grounds
Now that those caveats are out of the way, there have been some spectacular dinosaur egg discoveries over the last few decades. The most prominent is "Egg Mountain" in Nebraska, which has yielded not only intact, fossilized dinosaur eggs, but babies, juveniles and adults as well--all of them specimens of the late Cretaceous duck-billed dinosaur Maiasaura. Most important, from the perspective of dinosaur egg and nesting experts, is that the clutches of eggs in this Maiasaura nesting ground can be associated with the specific dinosaur that laid them, a frustratingly rare event. Similar, though not as spectacular, fossil finds have been made in central Asia, which has allowed paleontologists to link various eggs with dinosaurs as diverse as Troodon, Oviraptor and Protoceratops, and elsewhere in North America (most notably Hypacrosaurus).
These nesting grounds account for only a handful of the over 200 fossil sites worldwide that have yielded remnants of (mostly partial and unidentified) dinosaur eggs. More typical is a recent discovery in India, where a group of amateur fossil hunters stumbled across a hundred or so large, well-preserved dinosaur eggs. Because of the lack of accompanying fossil bones, though, the best experts can guess is that the eggs were laid by sauropods, and even this counts as a "probably" rather than a "definitely."
At this point you may ask: given the existence of intact, fossilized dinosaur eggs, can't we look inside them for fossilized embryos? Once again, matters aren't that simple: in the vast majority of cases, microscopic cracks in buried dinosaur eggs have long since been infiltrated by bacteria or parasites, which consume the embryo within. Even if the embryo somehow manages to avoid total destruction, only a few scattered bones may remain, which are virtually impossible to piece together into a convincing reconstruction of a fetal dinosaur.
Dinosaur Eggs and Dinosaur Survival
Up to now, we've only discussed the fossil remains of hundred-million-year-old dinosaur eggs, but how were these eggs laid, and how did they hatch, back when dinosaurs roamed the earth? Here, experts have more material to work with, mostly drawing on analogies with modern-day birds and reptiles. It's clear that dinosaurs laid their eggs in clutches of a few to a few dozen, often near bodies of water, and some species may have taken advantage of the warmth emitted by rotting vegetation to help incubate their brood (a surer bet than sitting on the eggs, like modern birds, the results of which you can imagine for a 20-ton Shantungosaurus female).
Until any evidence is found to the contrary, it seems probable that dinosaur hatchlings were "precocious"--that is, they were able to stand, feed, and run from predators as soon as they pecked (or head-butted) their way out of the shell. However, this wouldn't have prevented the vast majority of hatchlings from being eaten almost immediately by waiting predators; for most animals, reproduction is a matter of beating the odds, so all that would have been needed in order to propagate the species was for one or two baby Diplodocus (out of a clutch of two dozen) to scamper out of harm's way.
Much of the time, too, predators didn't wait for a given clutch of eggs to hatch before chowing down. There's some fossil evidence for prehistoric dinosaur egg breakfasts, with eggshells caved in from outside rather than pecked out from inside (though one can presume that the responsible carnivore didn't use a teaspoon, but its snout or a sharp hind claw). Considering the vulnerability of eggs and hatchlings, and the further perils awaiting juveniles, perhaps one or two eggs out of a hundred eventually produced a dinosaur of breeding age--the main evolutionary reason for laying those trillions of eggs in the first place.