Baryon asymmetry Baryon asymmetry refers to the observation that apparently there is matter in the Universe but not much antimatter. We don't see galaxies made of antimatter or observe gamma rays that would be produced if large chunks of antimatter would annihilate with matter. Hence at early times, when both were present, there must have been a little bit more matter than antimatter. This is quantified using the asymmetry parameter
$\eta = \frac{n_{baryon} - n_{antibaryon}}{n_{photon}}$
From cosmological measurements such as WMAP,
$\eta \approx (6 \pm 0.25) \times 10^{-10}$
However, the source of baryon asymmetry is said to be one of the Big Problems of Physics.
What is currently the state of the art regarding this puzzle? What's the best fit we can get from the Standard Model? What do we get from lattice simulations?
 A: To achieve a nonzero baryon asymmetry, one needs to satisfy the so-called Sakharov conditions:


*

*Baryon $B$ violation

*C-symmetry violation and CP-symmetry violation

*Interactions out of thermal equilibrium


If at least one of these "asymmetries" or "imbalances" is missing, the total $B$ of the Universe will remain zero.
The Standard Model preserves $B$ perturbatively (the baryon number is an accidental symmetry preserved by the renormalizable interactions), so it violates the first condition. In fact, while it violates C and CP, the violation of CP via the CKM matrix is too weak so even the second condition fails. The typical processes described by the Standard Model tend to be close to thermal equilibrium as well but the failure of one condition is enough so we don't need to be too specific about the last claim I made. The conclusion is clear: One needs models beyond the Standard Model to create a baryon asymmetry. This much has been known for decades.
However, the Standard Model contains some kind of generalized instantons, the sphalerons, that may convert the lepton number $L$ to $B$ and vice versa. So the first condition, $B$ violation, may be replaced by a combination of SM sphalerons and $L$ violation.
That's why the models producing the baryon asymmetry may be either the traditional "baryogenesis" in which the baryon number is clearly violated, or "leptogenesis" in which the lepton number is violated and the lepton asymmetry is later converted to a baryon asymmetry as well. One may discuss various thermal and non-thermal versions of these processes within various theories beyond the Standard Model such as the grand unified theories. The grand unified theories are rather characteristic for the literature because the characteristic scale of the physical phenomena responsible for leptogenesis or baryogenesis is usually assumed to be close to the GUT scale, up to 15 orders of magnitude above the LHC energies.
The literature on leptogenesis and baryogenesis is comparably large. For example, both words appear 700+ times in the titles, see

http://inspirehep.net/search?ln=en&p=find+title+baryogenesis&f=&action_search=Search
http://inspirehep.net/search?ln=en&p=find+title+leptogenesis&f=&action_search=Search

I invite you to check some of these papers. There's of course no universally acceptable, unambiguously superior model of leptogenesis or baryogenesis at this moment because there's also no clearly preferred model beyond the Standard Model. Physicists just don't know what the right mechanism producing the baryon asymmetry is. However, the field is sufficiently advanced that newly proposed models of beyond-the-Standard-Model physics are routinely tested on whether or not they provide us with a viable mechanism for baryogenesis or leptogenesis. 
A viable form of leptogenesis or baryogenesis is usually demanded together with a realistic enough implementation of cosmic inflation and with other cosmological constraints associated with high enough energy scales (e.g. the absence of the moduli problem etc.).
A: The only source of asymmetry in the Standard Model is from CP violation, and although there is CP violation in the Standard Model it is not large enough to account for the observed asymmetry. It's expected that the asymmetry will be explained by some extension to the standard model, but at the moment we don't know which, if any, of the suggested extensions is the culprit.
A: What if the gravitational force between matter and anti-matter is repulsive rather than attractive (an experiment is underway at CERN to answer this question)? Then the anti-matter galaxies would all be bound by gravity into super clusters (just like the matter galaxies) and we would not see any collisions between matter and anti-matter galaxies because their respective super clusters would be repelling each other. Furthermore, the space between super clusters would have been voided by a combination of annihilation between matter and anti-matter and the repulsive gravitational interaction long ago, allaying the concerns expressed above by  @anna v. The observed asymmetry could be an artifact of our limited power of observation (of anti-matter signatures) outside our local super cluster. Could this also explain dark energy? 
For this to be compatible with the observed distribution of dark matter in the large scale universe (connected filaments separated by voids with ordinary matter super clusters arrayed like beads on a string), it would be necessary for dark matter particles (whatever they are) to be their own anti-particles (like photons) and for them to attract both matter and anti-matter of the ordinary type via gravity.   
A: *

*Our Big Bang produced only matter. 2. Big Bangs can produce either only matter or only antimatter. 3. Many universes (only two will not do) were formed. One half of them contain only matter. The other half contains only antimatter. The formation of universes will continue. It has no known beginning nor a known end.

