Vortices as topological defects exist on all scales of Nature from cosmic strings to a dense quark matter. In macroscopic quantum systems like superconductors, superfluids and Bose-Einstein condensates the vortices are of special interest. There, they are quantized having “normal” cores where the superfluid order parameter is zero and its phase usually changes by 2π in a circle around the core. In type-II superconductors, which are all superconducting materials suitable for energy applications, an external magnetic field penetrates into the material as suggested by Abrikosov in his seminal paper , i.e. via vortices of supercurrents with a core of the size ξ, the superconducting coherence length and decaying on a distance λ, the penetration depth. The circulating supercurrents induce magnetic fields with a single flux quantum equal to the ratio of the Planck constant and two electron charges (h/2e). In bulk samples the vortex-vortex repulsion leads to formation of the Abrikosov vortex lattice. Pinning of the vortices is a crucial physical and technological issue to assure high critical currents for practical applications.
Progress in nanotechnology has raised an interest in mesoscopic superconductors which could be applied in future cryoelectronics and quantum computers. Vortices in mesoscopic superconductors behave dramatically different from the vortex matter in macroscopic samples. Why? Because the size of the object becomes comparable with important length scales of vortices. Indeed, ξ and λ have dimensions from unity to hundreds of nanometers. Then, the vortex-vortex interaction interferes with strong spatial confinement and the superconductor can bear only few flux quanta till it is destroyed by a magnetic field. Multivortex states arranged in the mesoscopic disks but also giant vortices have been proposed, see for example Refs.  and  explaining the Hall magnetometry measurements. Importantly, direct vortex imaging experiments on mesoscopic samples are available as well. I. V. Grigorieva et al.  have studied the vortex configurations via Bitter decoration on Nb disks with diameters of several micrometers. The vortices arranged in concentric vortex shells with “magic” numbers of vortices corresponding to opening a new shell in an increased magnetic field. Such a behaviour is reminiscent of shells in atoms but no “Hund's rules” are known for those “artificial atoms made from Cooper pairs rather than electrons” . When strong pinning of vortices is technologically introduced in such Nb mesoscopic disks  some vortices imaged by the Bitter decoration become much larger. They can be either giant vortices (circular symmetric states with a single core and fixed angular momentum carrying multiple flux quanta) or dense multicore vortex clusters. The Bitter technique which is only sensitive to vortex supercurrent/penetration depth has no possibility to distinguish these two states.
Recently, scanning tunnelling microscope (STM) experiments on vortices in nanocrystals have been published. STM is a suitable technique because it is sensing directly the vortex core. In their ultrahigh vacuum subkelvin STM experiment T. Cren et al.  have prepared Pb nanoislands of round or oval shape on a Si substrate. The nanocrystals had diameters between 80 to 140 nm and a thickness between 2.3 and 2.8 nm, i.e. less than 10 atomic layers of Pb. The effective coherence length (in mesoscopic samples ξ and λ are affected by the sample dimensions and the short electron's mean free path) ξeff became 25 nm securing strong confinement conditions for the vortices. In increasing magnetic fields vortex imaging on nanoislands shows a gradual penetration of single vortex, followed by other vortices with their maximum number equal to 3 or 4 depending on the size of the nanoisland. The multiple vortex configurations of both types have been directly observed. First, a very dense vortex cluster of two or even three vortices with distinct cores was seen followed by a formation of a giant vortex with a single core at even higher field strengths. The transition between the vortex cluster and the giant vortex seems to depend on the maximum size of the nanoisland. Certainly, further STM experiments on the samples with different spatial confinements will bring more light on this interesting subject.
Furthermore, symmetry confinement of vortices is another aspect in nanoscaled superconductors. The question arises what will happen if three vortices are to be placed in a square, or two vortices in a triangle? Chibotaru et al. have calculated such symmetry-induced formations of vortices in mesoscopic superconducting squares  and triangles  using the linearized Ginzburg-Landau (GL) equation. The results have shown that additional vortex-antivortex pairs nucleate spontaneously to preserve the symmetry of the sample. In a square, for example, the introduction of three flux quanta is realized by the appearance of four single vortices at the corners and one antivortex in the centre.
In most of the works on the spatial and symmetry confined vortices published so far the phenomenological GL approach which is valid only near the superconducting critical temperature has been used. However, the situations considered above are strictly speaking beyond the domain of the applicability of the GL theory. The paper by Chibotaru et al. in the present volume  presents a realistic microscopic model of the vortex configurations in a square-shaped nanosuperconductor. The model applicable at any temperature down to T = 0 K is based on the Bogolubov-deGennes theory. The authors consider strong spatial confinement where ξ is only five times smaller than the size of the square (a = 15 nm). The results for the vorticities from 0 to 3 in increasing magnetic fields have revealed a gradual penetration of a single vortex, then giant vortex with two flux quanta and finally four vortices with one antivortex, respectively. Some of the results reproduce the previous findings from the GL calculations but some are completely new. For example asymmetry conserving transition from a giant vortex with two flux quanta to a multiple vortex state with four vortices and one giant antivortex with two flux quanta is expected upon decreasing temperature.
In Fig. 1 some of the main results are presented: The order parameter shows oscillations on a length scale of the Fermi wavelength. In contrast to the GL model the single vortices are dispatched towards the edges of the square and not towards the corners. The macroscopic theory also predicts a sufficient vortex–antivor-tex separation as well as large amplitude of the order parameter in the region between vortex and antivortex. Since all those features should be observable by ultralow temperature STM experiments once samples with low symmetry (squares, triangles) will be ready for tunneling experiments, we can look forward to further interesting nanophysics.