  Spin-$S$ Heisenberg quantum antiferromagnets on the Kagome lattice offer,
when placed in a magnetic field, a fantastic playground to observe exotic
phases of matter with (magnetic analogs of) superfluid, charge, bond or nematic
orders, or a coexistence of several of the latter. In this context, we have
obtained the (zero temperature) phase diagrams up to $S=2$ directly in the
thermodynamic limit thanks to infinite Projected Entangled Pair States (iPEPS),
a tensor network numerical tool. We find incompressible phases characterized by
a magnetization plateau vs field and stabilized by spontaneous breaking of
point group or lattice translation symmetry(ies). The nature of such phases may
be semi-classical, as the plateaus at $\frac{1}{3}$th, $(1-\frac{2}{9S})$th and
$(1-\frac{1}{9S})$th of the saturated magnetization (the latter followed by a
macroscopic magnetization jump), or fully quantum as the spin-$\frac{1}{2}$
$\frac{1}{9}$-plateau exhibiting coexistence of charge and bond orders. Upon
restoration of the spin rotation $U(1)$ symmetry a finite compressibility
appears, although lattice symmetry breaking persists. For integer spin values
we also identify spin gapped phases at low enough field, such as the $S=2$
(topologically trivial) spin liquid with no symmetry breaking, neither spin nor
lattice.
