  The iron-based LaFeAsO$_{1-x}$F$_x$ recently discovered by Hosono's group is
a fresh theoretical challenge as a new class of high-temperature
superconductors. Here we describe the electronic structure of the material and
the mechanism of superconductivity. We start with constructing a tight-binding
model in terms of the maximally localized Wannier orbitals from a
first-principles electronic structure calculation, which has turned out to
involve all the five Fe 3d bands. This is used to calculate the spin and charge
susceptibilities with the random phase approximation. The spin susceptibility
has peaks around ${\Vec k} = (\pi, 0), (0, \pi)$ arising from a nesting across
disconnected Fermi surface pockets. We have then plugged the susceptibilities
into the linearised Eliashberg equation. For the doping concentration $x =
0.1$, we obtain an unconventional s-wave pairing, which is roughly an extended
s in that the gap changes sign between the Fermi pockets, but the gap function
is actually a 5$\times$5 matrix. Its experimental implications are also
discussed.
