Three-hadron spectroscopy is a key frontier in our understanding of the hadron spectrum. In recent years, significant formal and numerical advances have paved the way for studying three-hadron processes directly from lattice QCD, with outstanding applications including the Roper resonance and the doubly charmed tetraquark. This requires theoretical frameworks that relate finite-volume energies to infinite-volume three-particle scattering amplitudes. In this contribution, I discuss recent progress in formulating such frameworks for generic three-hadron systems, and present numerical results for three-meson systems at maximal isospin with physical quark masses, as well as our recent investigation of the three-body dynamics of the doubly charmed tetraquark, $T_{\rm cc}$.