  Recent technological advances in controlling materials have developed methods
to produce idealized two-dimensional (2D) electron systems such as
heterogeneous interfaces, molecular-beam-epitaxy (MBE) grown atomic layers,
exfoliated thin flakes and field-effect devices. These 2D electron systems are
highly-crystalline with less disorder in common, some of which indeed show
sheet resistance more than one order of magnitude lower even in atomic layers
or single layers than that of conventional amorphous/granular thin films. Here,
we present a review on the recent developments of highly-crystalline 2D
superconductors and a series of unprecedented physical properties discovered in
these systems. In particular, we highlight the quantum metallic state (or
possible metallic ground state), the quantum Griffiths phase in out-of-plane
magnetic fields, and the superconducting state maintained in anomalously large
in-plane magnetic fields, which were observed in exfoliated 2D materials,
MBE-grown atomic-layer thin films and electric-double-layer (ion-gated)
interfaces. These phenomena are discussed on the basis of weakened disorder
and/or broken spatial inversion symmetry. These novel aspects suggest that
highly-crystalline 2D systems are promising platforms for exploring new quantum
physics and superconductors.
