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
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.).