  The novel spin-triplet superconductor candidate UTe2 was discovered only
recently at the end of 2018 and attracted enormous attention. We review key
experimental and theoretical progress which has been achieved in different
laboratories. UTe2 is a heavy-fermion paramagnet, but right after the discovery
of superconductivity it has been expected to be close to a ferromagnetic
instability showing many similarities to the U-based ferromagnetic
superconductors, URhGe and UCoGe. The competition between different types of
magnetic interactions and the duality between the local and itinerant character
of the 5f Uranium electrons, as well as the shift of the U valence appear as
key parameters in the rich phase diagrams discovered recently under extreme
conditions like low temperature, high magnetic field, and pressure. We discuss
macroscopic and microscopic experiments at low temperature to clarify the
normal phase properties at ambient pressure. Special attention will be given to
the occurrence of a metamagnetic transition at Hm = 35 T for a magnetic field
applied along the hard magnetic axis b. Adding external pressure leads to
strong changes in the magnetic and electronic properties with a direct feedback
on superconductivity. Attention will be given on the possible evolution of the
Fermi surface as a function of magnetic field and pressure. Superconductivity
in UTe2 is extremely rich exhibiting various unconventional behaviors which
will be highlighted. It shows an exceptionally huge superconducting upper
critical field with a re-entrant behavior under magnetic field and the
occurrence of multiple superconducting phases in the temperature field,
pressure phase diagram. There is evidence for spin-triplet pairing. The
different theoretical approaches will be described. Notably we discuss that
UTe2 is a possible example for the realization of a fascinating topological
superconductor.
