I will review recent work on multi-gap topological states. These phases are characterized by topological structures that cannot be captured by advances in more conventional symmetry-based topological classifications schemes. Upon utilizing new insights into connections with general geometric identities I will elucidate the structure of these phases and highlight physical signatures in both equilibrium and out-of-equilibrium settings. These signatures are increasingly finding pathways to the laboratory in contexts that range from meta-materials and quantum simulators to electronic and phononic structures in real material candidates. As a highlight I will address how this understanding vice versa also relates to new takes on formulating novel quantum geometrical framework that can be probed by physical responses.