  New light, weakly-coupled particles are commonly invoked to address the
persistent $\sim 4\sigma$ anomaly in $(g-2)_\mu$ and serve as mediators between
dark and visible matter. If such particles couple predominantly to heavier
generations and decay invisibly, much of their best-motivated parameter space
is inaccessible with existing experimental techniques. In this paper, we
present a new fixed-target, missing-momentum search strategy to probe invisibly
decaying particles that couple preferentially to muons. In our setup, a
relativistic muon beam impinges on a thick active target. The signal consists
of events in which a muon loses a large fraction of its incident momentum
inside the target without initiating any detectable electromagnetic or hadronic
activity in downstream veto systems. We propose a two-phase experiment, M$^3$
(Muon Missing Momentum), based at Fermilab. Phase 1 with $\sim 10^{10}$ muons
on target can test the remaining parameter space for which light
invisibly-decaying particles can resolve the $(g-2)_\mu$ anomaly, while Phase 2
with $\sim 10^{13}$ muons on target can test much of the predictive parameter
space over which sub-GeV dark matter achieves freeze-out via muon-philic
forces, including gauged $U(1)_{L_\mu - L_\tau}$.
