Nilsequences, namely sequences of complex numbers arising naturally from translations on nilmanifolds, introduced by Bergelson, Host & Kra, appear in different guises in several areas of mathematics: Topological Dynamics (maximal nilfactors), Ergodic Theory (convergence of multiple ergodic averages), Additive Number Theory (solving linear equations in primes) and Additive Combinatorics (generalizations of Szemerédi's Theorem). A flexible framework to investigate nilsequences and related concepts was introduced by Camarena and Szegedy. The main object, a nilspace, is a (compact) space which satisfy some straightforward axioms. A fundamental result is the reprehensibility of a nilspace as an inverse limit of a nilmanifolds. In a work in progress with Freddie Manners and Péter Varjú we give a new proof of this fundamental result. The new tools allow us to derive generalizations of the Host-Kra-Maass Structure Theorem for topological dynamical systems of finite order.