  Although the pairing mechanism of the Fe-based superconductors (FeSCs) has
not yet been settled with a consensus, as to the pairing symmetry and the
superconducting (SC) gap function, the abundant majority of experiments are
supporting for the spin-singlet sign-changing s-wave SC gaps on multibands
($s^{\pm}$-wave state). This multiband $s^{\pm}$-wave state is a very unique
gap state {\it per se} and displays numerous unexpected novel SC properties
such as a strong reduction of the coherence peak, non-trivial impurity effects,
nodal-gap-like nuclear magnetic resonance (NMR) signals, various Volovik
effects in the specific heat (SH) and thermal conductivity, and anomalous
scaling behaviors with the SH jump and the condensation energy vs. $T_c$, etc.
In particular, many of these non-trivial SC properties can be easily mistaken
as evidence for a nodal gap state such as a d-wave gap. In this review, we
provide detailed explanations of theoretical principles for the various
non-trivial SC properties of the $s^{\pm}$-wave pairing state, and then
critically compare the theoretical predictions with the experiments of the
FeSCs. This will provide a pedagogical overview of how much we can coherently
understand the wide range of different experiments of the FeSCs within the
$s^{\pm}$-wave gap model.
