Analysis of geometrically and topologically nontrivial nanoarchitectures at the nanoscale is of immense importance for multidisciplinary scope of electronics, spintronics, magnetism, optics, optoelectronics, thermoelectrics and quantum computing. Advances of high-tech fabrication techniques have allowed for generating topologically nontrivial manifolds with man-made space metrics, which determine electronic, optical, magnetic and transport properties of such objects and novel potentialities of nanodevices due to their unique geometry and topology [1]. The physics of quantum rings is overviewed from basic concepts rooted in the quantum-mechanical paradigm—via unprecedented challenges brilliantly overcome by both theory and experiment—to promising application perspectives [2]. Symbiosis of the geometric potential and an inhomogeneous twist renders an observation of the topology effect on the electron ground-state energy in microscale Möbius strips into the realm of experimental verification. A delocalization-to-localization transition for the electron ground state is unveiled in inhomogeneous Möbius strips [3]. The pattern of superconductor vortices in a micro- or nanoarchitecture represents therefore an important study case for arranging topological defects in confined geometries. The roll-up fabrication methods have provided qualitatively novel curved superconductor micro- and nanoarchitectures, e.g., nanostructured microtubes [4] and microhelices [5]. Vortex dynamics in open superconductor microtubes in the presence of a transport current are influenced by the interplay between the scalar potential and the inhomogeneous magnetic field component normal to the surface. Using the inhomogeneous transport current allows one to control the branching and to reduce the average number of vortices in the tube [6]. A Möbius ring resonator [7] and a rolled-up asymmetric microcavity [8] are representative examples, which give rise to fascinating topological effects by virtue of Möbiosity, spin-orbit coupling and non-Abelianism. The cone-like rolled-up asymmetric microcavities provide a platform to realize spin–orbit coupling of light for the examination of nontrivial topological effects in the context of a non-Abelian evolution [8]. In asymmetric microcavities, the geometric phase is directly measured by monitoring the polarization tilt angles, while the eccentricities indicate the mode conversion between the right and left circular bases. Those findings imply promising applications by manipulating photons in on-chip quantum devices. Effects of nontrivial geometry and topology for tailoring physical properties of phonons in novel micro- and nanoarchitectures embrace reduction of the lattice thermal conductivity in one-dimensional quantum-dot superlattices [9] and cross-section modulated quantum wires [10] and a prominent impact of the number of shells on the phonon dispersion, phonon group velocity dispersion and the density of states of phonons as well as on the lattice transport in multishell nanotubes [11].