New research has found that the neutrino has a non-zero mass - but this requires a modification to the Standard Model of particle physics. The non-zero mass also means that neutrinos cannot travel at light-speed as photons do.
“Although neutrinos were long believed to be massless, it is now known that there are three discrete neutrino masses with different tiny values, but they don't correspond uniquely to the three flavors. A neutrino created with a specific flavor is in an associated specific quantum superposition of all three mass states”
Additionally, to acquire mass, all fermions (including neutrinos) of both 'handedness' must interact with the Higgs boson.
“In the Standard Model of particle physics, fermions only have mass because of interactions with the Higgs field. These interactions involve both 'left-handed' and 'right-handed' versions of the fermion. However, only left-handed neutrinos have been observed so far.”
Further, the mass that neutrinos do have has not yet been measured :
“It might seem absurd that physicists don't know how much neutrinos weigh, given that the universe contains more of them than any other type of matter particle. However, it's impossible to capture neutrinos, so there's no simple way to measure their mass.”
Source: Science, Vol. 356, Issue 6345, pp. 1322-1326, 2017
The mass which they have must (according to the current 'Standard Model') be acquired via the Higgs boson - but there are theoretical conflicts regarding neutrinos :
“The Higgs boson is responsible for giving mass to a certain class of particles that feature in the theory of the so-called electroweak interaction: a unified description of two of the four fundamental forces, electromagnetism and the nuclear weak force. Because the weak force underlies the beta decay process in which neutrinos are formed, neutrinos are implicated in this “Higgs mechanism.” In short, the non-zero mass of the three types of neutrino and the oscillations that cause switches between them can’t be fully squared with the Higgs mechanism, and seem to demand some kind of physics beyond the Standard Model.”
Also See: Radioactive decay neutrino anomaly