  Unconventional superconductivity (USC) in a highly correlated kagome system
has been theoretically proposed for years, yet the experimental realization is
hard to achieve. The recently discovered vanadium-based kagome materials, which
exhibit both superconductivity and charge density wave (CDW) orders, are
nonmagnetic and weakly correlated, thus unlikely host USC as theories proposed.
Here we report the discovery of a chromium-based kagome metal, CsCr$_3$Sb$_5$,
which is contrastingly characterised by strong electron correlations,
frustrated magnetism, and characteristic flat bands close to the Fermi level.
Under ambient pressure, it undergoes a concurrent structural and magnetic phase
transition at 55 K, accompanying with a stripe-like $4a_0$ structural
modulation. At high pressure, the phase transition evolves into two
transitions, probably associated with CDW and antiferromagnetic
spin-density-wave orderings, respectively. These density-wave (DW)-like orders
are gradually suppressed with pressure and, remarkably, a superconducting dome
emerges at 3.65-8.0 GPa. The maximum of the superconducting transition
temperature, $T_\mathrm{c}^{\mathrm{max}}=$ 6.4 K, appears when the DW-like
orders are completely suppressed at 4.2 GPa, and the normal state exhibits a
non-Fermi-liquid behaviour, reminiscent of USC and quantum criticality in
iron-based superconductors. Our work offers an unprecedented platform for
investigating possible USC in a correlated kagome system.
