Why do or don't neutrinos have antiparticles? This was inspired by this question. According to Wikipedia, a Majorana neutrino must be its own antiparticle, while a Dirac neutrino cannot be its own antiparticle. Why is this true?
 A: Because the spinor of the Majorana neutrino is an eigenstate of the charge conjugation operator. This is different from the case of a Dirac spinor that will change under the effect of the same operator.
A: It's often stated that if neutrinos are Dirac fermions then they're distinct from their antiparticles, and if they're Majorana fermions then they're the same as their antiparticles. Whether or not this is true depends on how do you define both "Dirac and Majorana fermions" and also "antiparticle", but in my opinion the statement is misleading according to its most natural interpretation.
When people talk about neutrinos being Dirac or Majorana fermions, they can mean at least three different incompatible things:


*

*A fermion bispinor field configuration $\Psi(x)$ is sometimes defined to be a "Majorana field" if it is required to be left invariant by (not just to be an eigenstate of) the charge conjugation operator. In this case the field only has one independent Weyl field and two independent spin indices. $\Psi(x)$ is a "Dirac field" if it is not required to be invariant under charge conjugation, so that it has four independent spin components. Note that this is solely a statement about the field configuration, not about any Lagrangian that might describe the field's dynamics. Whether a bispinor field is Dirac or Majorana is a mathematically well-defined question but has no physical content, because quantum fields are not directly observable. When we talk about antiparticles in this context, we really mean the charge conjugation operator, and whether or not it leaves the quantum field invariant.

*A particular formulation of a quantum field theory (i.e. a choice of field content and Lagrangian) is a formulation in terms of Dirac or Majorana fermions if the matter fields are expressed in terms of Dirac or Majorana bispinor fields, respectively. This distinction also has no physical content; indeed any theory of spin-1/2 fermions can be equivalently formulated in terms of either Dirac or Majorana bispinor fields, and either choice can be more convenient in different circumstances.

*A quantum field theory is sometimes described as a theory of "Dirac fermions" if its action has a continuous gauge or global symmetry that ensures the conservation of lepton number, and a theory of "Majorana fermions" if not. Unlike the first two pairs of definitions, this one has physically distinct consequences (e.g. the absence or presence of neutrinoless double beta decay). However, it's only loosely connected to the first two. (The loose connection is that it's mathematically simpler to formulate such a symmetry in terms of a Dirac bispinor field.) 
The third definition is the most common, but it's important to note that in the simplest formulation of "Majorana neutrinos" in beyond-Standard-Model physics, the neutrino fields are not Majorana bispinor fields, but instead are Dirac bispinor fields with mass terms that violate conservation of lepton number. So in my opinion, it's misleading to say that "Majorana neutrinos are their own antiparticles," since charge conjugation takes a Majorana neutrino field to a distinct field.
The motivation for that terminology is that operationally/experimentally, particles and antiparticles are characterized by the conservation of charge or ((particle number) - (antiparticle number)). Without such a conservation law, particles and antiparticles are difficult to distinguish in practice.
Nevertheless, I personally would say that it's correct to say that "there is no physical distinction between Majorana fermion particles and antiparticles," but it's incorrect to say that "a Majorana fermion is its own antiparticle." The distinction is subtle but important: the latter claim gives the misleading impression that Majorana fermion particles and antiparticles are both well-defined concepts that happen to 
coincide, while the former claim conveys that it just doesn't really make sense to talk about antiparticles at all in the context of Majorana fermions.
A: In crude terms I think it amounts to the following:
Consider, for instance, a (Dirac) fermion creation operator: $c_j^\dagger$. A Majorana fermion is somehow the "real" Part of a Dirac fermion:
$m_j = c_j + c_j^\dagger$ 
(conventions on normalization differs). Hence a Majorana fermion transforms into itself under charge conjugation.  
A: here is how I understood the answer. Imagine that we look at the electron in our reference frame, and find that it travels in z direction and has spin projection +1/2. This one we call right-handed electron. Since it is massive, there exist reference frames in which observers see it as left-handed electron. For example those observers that travel faster than the electron in the z-direction. Its charge is Lorentz invariant quantity, so everyone agrees that this is electron and not positron. Massive electron is then described by four degrees of freedom (4 basic spinors), left and right handed electron and left and right-handed positron; it is Dirac field.
Imagine now that we have left handed neutrino in our reference frame. Since we have lot of evidence that neutrino is massive, we can see that there are reference frames in which right handed neutrino is seen. The right handed neutrino is not observed in any experiment so far, and their mass could be very high. We only observed left handed neutrino and right handed antineutrino. So we can consistently demand that the observer in the other reference frame actually sees right handed antineutrino, without introducing right handed neutrino and left handed antineutrino. For charged particle this would not work. So we can describe neutrino by only two degrees of freedom; it is then Majorana neutrino. But the neutrinos are charged under lepton number symmetry which is global symmetry of the Standard Model. Being global means that it does not create dynamics, and there is no fundamental reason why it cannot be broken. If (local) gauge symmetry is explicitly broken then it leads to inconsistency of the theory, meaning that non-physical degrees of freedom could arise.
So Majorana particles are their own antiparticles only if we do not take into account lepton number symmetry. 
A: Neutrino definitely have anti-particles. Another question  whether an anti-particle differs from its "particle" or not. The charge conjugation operator may generally change a given "neutrino state" because a neutrino solution is not completely determined with just Dirac's equation.
