One important class of supramolecular materials is one-dimensionally elongated supramolecular polymers, in which monomers are associated by reversible intermolecular interactions, yielding a fibrous morphology. Unlike frequently reported conventional supramolecular polymers based on, for instance, host-guest interactions, those composed of one-dimensionally stacked π-conjugated molecules can be encoded with high degrees of internal order by cooperative association of the rigid aromatic monomers, endowing such supramolecular polymers with extraordinary properties and functionality. However, their internal order has not yet been exploited to manipulate the complex landscape of well-defined states of the supramolecular polymer backbone, which may induce new functionalities beyond the intrinsic properties of the backbones. This Account will focus on the inceptive phase of our research on supramolecular polymers with high degrees of internal order able to impart intrinsic curvature to their backbones. Initially, we developed a naphthalene molecule functionalized with barbituric acid, which forms uniform toroidal short fibers with diameters of approximately 16 nm via the formation of hydrogen-bonded cyclic hexamers (rosettes). As we thought the uniformity of the toroid size to arise from the intrinsic curvature generated upon stacking of the rosettes, we exploited this intrinsic curvature to design continuously curved extended supramolecular polymers by extension of such molecular π-systems. The intrinsic curvature produced by the monomers with more expanded π-systems indeed gave us access to higher-order structures (topologies) ranging from randomly folded to helically folded coils in extended supramolecular polymers. We will discuss the kinetic aspects of the generation of intrinsic curvature for topology control, including the formation of toroidal structures resulting from ring-closing processes. For extended supramolecular polymers with well-defined topologies, we will discuss manipulation of a complex landscape of well-defined states by external stimuli. The incorporation of a photoresponsive azobenzene chromophore in the original naphthalene molecular scaffold allowed us to reversibly destroy or recover the curvature of the main chain through trans- cis photoisomerization. By means of this photocontrollable curvature, we have demonstrated light-induced unfolding of helically folded structures into entirely stretched structures. Furthermore, a direct extension of the π-conjugated core provided us with access to unprecedented supramolecular polymers with emergent time-dependent topology transitions. Molecules with a naphthalene core conjugated with two phenylene units kinetically afforded supramolecular polymers that consist of helically folded and misfolded domains. Upon aging the supramolecular polymer solution, we observed spontaneous folding of the misfolded domains in a time scale of days, eventually obtaining a supramolecular polymer topology analogous to the tertiary structure of proteins. These supramolecular polymers with unrivaled and active topologies provide new prospects for supramolecular polymers as one-dimensional nanomaterials.