  Discovery of unexpected properties of the Higgs boson offers an intriguing
opportunity of shedding light on some of the most profound puzzles in particle
physics. The Beyond Standard Model (BSM) decays of the Higgs boson could reveal
new physics in a direct manner. Future electron-positron lepton colliders
operating as Higgs factories, including CEPC, FCC-ee and ILC, with the
advantages of a clean collider environment and large statistics, could greatly
enhance the sensitivity in searching for these BSM decays. In this work, we
perform a general study of Higgs exotic decays at future $e^+e^-$ lepton
colliders, focusing on the Higgs decays with hadronic final states and/or
missing energy, which are very challenging for the High-Luminosity program of
the Large Hadron Collider (HL-LHC). We show that with simple selection cuts,
$O(10^{-3}\sim10^{-5})$ limits on the Higgs exotic decay branching fractions
can be achieved using the leptonic decaying spectator $Z$ boson in the
associated production mode $e^+e^-\rightarrow Z H$. We further discuss the
interplay between the detector performance and Higgs exotic decay, and other
possibilities of exotic decays. Our work is a first step in a comprehensive
study of Higgs exotic decays at future lepton colliders, which is a key
ingredient of Higgs physics that deserves further investigation.
