I review recent lattice results relevant to models of compositeness for physics beyond the Standard Model, with a strong focus on work performed by the TELOS Collaboration on $Sp(4)$ gauge theories with matter in multiple representations. After motivating the Composite Higgs framework and detailing the theoretical underpinning of the $Sp(4)$ gauge model, I set the scene with a discussion of recent calculations of the flavoured meson spectrum, including continuum-limit results for both the fundamental and antisymmetric fermionic sectors. I further discuss the first application of the Hansen-Lupo-Tantalo spectral-density method to the multiple representation theory and the resulting determination of chimera baryon masses and matrix elements. I then review the determination of isosinglet states and report on non-perturbative calculations of dark pion scattering cross sections that are relevant to strongly interacting massive particle (SIMP) scenarios of dark matter. Finally, I discuss the thermodynamics of the pure $Sp(4)$ Yang--Mills theory, which is relevant to the production of gravitational waves from early-universe first-order phase transitions.