Are neutrinos Majorana particles? That is, are they identical to their anti-particles? (Any results of double beta decay experiments?)
 A: This is just a clarification regarding the see-saw. The see-saw is inside a particular model--- an SO(10) GUT. It is interesting, but Majorana neutrino masses of the right order of magnitude do not require an SO(10) GUT, or any other type of GUT, in order to work. All they require is that there is some new interactions of a generic nature at near the GUT scale. This interaction generically produces a nonrenormalizable term in the low-energy action of the form
$$ H^a H^b \epsilon_{\alpha\beta} L^\alpha_a L^\beta_b $$
plus the complex conjugate, Where H is the Higgs field, L is the left-handed lepton doublet, and $\epsilon$ is the Lorentz undotted-index epsilon-tensor, the latin indices are weak SU(2) indices, and the greek indices are Lorentz indices. This term, when you reduce it into components by the Higgs mechanism, is a Majorana neutrino mass, written in Weyl form. The magnitude of the mass is the coefficient of this term times the Higgs VEV squared, divided by the non-renormalizability scale.
The interpretation of this term is that it is a  simultaneous scattering of a neutrino from two Higgs bosons in the condensate. This double-scattering is suppressed by the scale at which the scatterings are not simultaneous, because simultaneous fermi-fermi-scalar-scalar interactions (unlike simultaneous four-scalar interactions) are not renormalizable.
The SO(10) GUT see-saw mechanism is just one way of producing such a term, which is nice because it is inside an explicit model where the high-energy theory is again renormalizable. But this is not really necessary, because the GUT scale is so close to the quantum gravity scale. The resolution of the double-Higgs scattering could be directly with a string intermediate, or with twenty different intermediate Fermions, or whatever you want. It falls out of any theory which changes things at the GUT scale.
If the neutrinos are not Majorana, they would need a partner to produce a mass. This partner would have to have no standard model gauge charges, and could itself acquire a huge Majorana mass in principle. To say that it doesn't requires fine tuning. Further, the interaction between this sterile neutrino and the ordinary neutrino would not be naturally suppressed by their non-renormalizability, as the Majorana neutrino masses are, and would require further extraordinary fine tuning.
It is things like this that make it certain for me that the Neutrino is Majorana. Barring any contradicting experimental data, this should be considered dead certain.
A: Neutrinoless double beta decay, a sure sign of Majorana-ness of the neutrino, was claimed to be observed by Klapdor-Kleingrothaus et al. (with 2-3 sigma), see this paper. However, their results are far from being accepted by the community.
But it is certainly not excluded that neutrinos are Majorana particles.
A: Great question. The experimental situation remains inconclusive. However, theoretically, there exists a damn good reason to think that the neutrinos have Majorana masses - and, consequently, the double beta decays should be possible. It's called the seesaw mechanism.
The mechanism is justified by an intriguing observation:
$$ m_{\nu}:m_{Higgs} \approx m_{Higgs}:m_{GUT}$$
Both ratios are about $10^{-14}$: note that $m_\nu, m_{Higgs}, m_{GUT}$ are about $10^{-12},10^2,10^{16}$ GeV. The light, millielectronvolt-scaled neutrinos, seem to be on the other side of this geometric sequence.
This fact can be explained by a rather robust mechanism. The light neutrino masses - written as a $3\times 3$ mass matrix - may be generated as
$$ m = -m_D M_R^{-1} m_D^T. $$
The Dirac masses $m_D$, linking the known left-handed neutrinos with the heavy, unknown, right-handed neutrinos, are comparable to the Higgs mass - the electroweak scale. Meanwhile, the Majorana masses for the right-handed neutrinos, $M_R$, are comparable to the GUT scale. Right-handed neutrinos are automatically predicted e.g. as the "invisible" 16th component of the 16-dimensional spinor multiplet in $SO(10)$ grand unified theories, among others.
The formula above automatically produces the "seesaw": the resulting masses $m$ are as many times smaller than $m_D$ as $m_D$ is smaller than $m_R$.
It's very natural for the right-handed neutrinos to acquire Majorana masses: they're pretty much neutral and nothing prevents them from being coupled in pairs. The Dirac masses comparable to the electroweak scale may also be justified.
As a consequence, one automatically obtains the small masses for the known, left-handed neutrinos: and they're inevitably Majorana masses, too. They can't be Dirac masses simply because Dirac masses would require the right-handed parts of the particles to be as light as the left-handed one. But the right-handed neutrinos are superheavy and may be neglected.
The seesaw formula may be obtained by diagonalizing a matrix $((0,m_D),(m_D,M_R))$. One eigenvalue is inevitably close to $M_R$ (which are really three eigenvalues); the other is much smaller than $m_D$. Alternatively, one may view $1/M_R$ in the formula as a suppression factor from more general higher-dimensional operators.
It should be emphasized that the Dirac masses are used in the seesaw formula. However, because they're much smaller than the right-handed Majorana masses, the Dirac masses don't dictate the basic character of the fermions.
Of course, there may be surprises, for example both Majorana and Dirac masses may occur at low energies, too. That would require some mixing of the three generations of the neutrinos - a scenario that is problematic for many reasons but it is not excluded.
In the text above, I assumed that the reader understands that the fields needed to describe a Weyl fermion and a Majorana fermion are the same fields: a 2-component complex spinor. Automatically, its Hermitean conjugate (one with the opposite chirality) also exists as an operator on the Hilbert space.
Whether it is more natural to call this field "Weyl" or "Majorana" depends on the mass terms (or the interaction terms). Those that dominate decide about the name. If the Majorana mass terms are much larger than the Dirac masses, the particles are essentially Majorana particles, and the lepton number violation may be substantial while parity is kind of conserved. If the Dirac masses are much larger, the parity is qualitatively preserved - but one needs a doubled number of excitations (left-handed and right-handed particles of the same or similar mass).
There can't be any mass terms that would both preserve the lepton number (distinguished particles and particles) and that would also keep the neutrinos purely left-handed.
A: If neutrinos were Majorana particles lepton number conservation would not work. In http://hitoshi.berkeley.edu/neutrino/neutrino4.html they summarize:
Implications of neutrino mass:
Neutrinos are found to have mass, but the mass is extremely tiny, at least million times lighter than the lighest elementary particle: electron. How do we need to change the Standard Model to explain the neutrino mass? Some argue that our spacetime has unseen spatial dimensions, and we are stuck on three-dimensional "sheets". Other argue that we need to abandon the sacred distinction between matter and anti-matter. 
Majorana neutrinos would require to abandon the distinction between matter and antimatter, and change the Standard Model. 
