There are various theories about the origin of the universe. The one that enjoys the most support today in the scientific establishment is the Big Bang Theory, which states that the universe began about 12 billion years ago as an infinitesimally small capsule of energy that rapidly expanded and formed everything we now see. The image shows NASA's idea of what the early universe may have looked like according to this theory. The following are some of the problems with this theory:-
Baryon asymmetry: According to the Big Bang Theory, all matter today has arisen from energy. When energy gets converted to matter, it forms equal amounts of matter and anti-matter. But as far as we know, there is much more matter in the universe today than anti-matter. This is a mystery, and is being researched actively by scientists today.
Lumpiness: The known laws of physics, when applied to the Big Bang Model, predict that the universe should be just an expanding cloud of gas. However, the universe has a lot of "lumps". Stars and planets are concentrations of matter in a space that is almost empty. Stars are organized into galaxies, and galaxies themselves are organized into clusters and super-clusters. The Sloan Great Wall super-cluster of galaxies has a length of nearly 1% of the known universe. Such lumpiness is totally inconsistent with the big bang theory. According to the Big Bang Theory, star clusters formed at different times and places, so they should show a lot of variation, but their observed composition (the proportion of small, medium and big stars) is nearly the same in all the star clusters that have been studied (see T. Abel, G.L. Bryan, and M.L. Norman, "The formation of the first star in the universe", Science 295:93-98 (2002)).
Dark matter: Not only should galaxy clusters not form, but if they formed, they should have been pulled apart by now due to their relative motion and the expansion of the universe. But they are still together - the gravitation between galaxies in clusters has been unexpectedly strong (if the Big Bang really took place). Huge amounts of invisible "dark matter" have been invoked to explain this. "Dark matter" is a classic example of how enthusiasts of a theory can invent non-observed entities to deal with observations that otherwise don't agree with their theory.
There is an interesting implication of dark matter: If 80% of the universe is "dark", and thus unknown, no current theory can claim to "explain" the universe!
Isotropy: The Cosmic Microwave Background Radiation (supposedly the after-glow of the Big Bang) supposedly comes from a period after the big bang when everything was supposed to be isotropic (same in every direction), so it should be isotropic, but it is not. It suggests a cosmic south pole, north pole and equator. Dr Max Tegmark of the University of Pennsylvania, who processed the data, called the result bizarre, unexpected and without explanation (he is quoted in the BBC News report: "Map Reveals Strange Cosmos"). In the Sloan Digital Sky Survey, the galaxies appear to be in concentric shells centered on our galaxy and they are far more dense close in than farther out. This is in total opposition to the predictions of the big bang theory.
Fine tuning: If the initial expansion claimed by the Big Bang was a little too slow, gravity would have overpowered the universe and contracted it back to a capsule within a fraction of a second. If the expansion was a little too fast, lumps could never form, and thus we wouldn't exist. The "window" of variation permissible is 1 part in 10 raised to 60. To get a feel for this, imagine two numbers: 1 and 1.0000..(59 zeroes)..1. The expansion rate should be between these two values. Our very existence, assuming a Big Bang scenario thus implies a very precise expansion rate, and it is unreasonable that a random explosion would have just the right expansion rate. [See Alan Guth, "Inflationary Universe: A Possible Solution to the Horizon and Flatness Problems" Physical Review D 23 (2):347-356 (1981)]
The origin of the "cosmic egg": According to the Big Bang Theory, the big bang is a "singularity" - a moment where the laws of physics do not apply; in other words, it is a MIRACLE. Thus, Big Bang proponents are guilty of double standards when they frown on the Biblical creation account, which involves miracles.
Abundance of the elements: According to the Big Bang Theory, the current abundance of the 92 elements requires many generations of massive stars recycling elements again and again. But such stars would become black holes - then the number of black holes should have been comparable to the number of living stars, but there are only a few dozen candidates for black holes in our galaxy (as against 100 billion stars).
For a more detailed analysis of the flaws of the Big Bang Theory, see:-
New Scientist published an Open Letter to the Scientific Community written by secular scientists unhappy about the strong arm tactics used by Big Bang proponents to silence dissent. Other secular scientists have also expressed their skepticism.