Rare and very rare decays at the LHCb experiment

Rare and very rare decays of third-generation particles, including $b$-hadrons and $\tau$ leptons, provide sensitive probes of physics beyond the Standard Model (SM). Unlike direct searches limited by collider energies, they probe new physics at much higher energy scales. Many of these decays have SM-predicted branching fractions below the sensitivity of current detectors. These proceedings report on recent LHCb searches, including several first searches and results setting the most stringent limits to date. In particular, searches for $b \to s \tau^+\tau^-$, $b \to s \tau^\pm e^\mp$, $b \to s \mu^\pm e^\mp$, and $\tau^- \to \mu^-\mu^+\mu^-$ are presented, alongside searches for lepton-number-violating processes and loop-suppressed annihilation decays.