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In recent years, extensive research has been conducted into the design of high-power-density rotating machines. Due to their unrivalled ability to carry large current densities under high magnetic fields, superconductors are among the best candidates for these future machines. To achieve unprecedented power densities, most envisioned designs propose reducing (or entirely eliminating) ferromagnetic components, which allows decreasing the machine weight and operating at fields exceeding the saturation magnetisation of ferromagnetic materials. As a downside, significant stray magnetic fields are generated outside the machine and must be attenuated to protect human operators and sensitive equipment nearby. Therefore, methods to reduce efficiently these inhomogeneous, low-frequency, high-magnitude stray magnetic fields extending over large regions must be studied. This thesis investigates passive high-temperature superconducting (HTS) screens or shields, exploiting the permanent currents naturally induced in superconductors when subjected to a varying applied field. The goal is to explore how to design HTS magnetic screens having the potential to be intrinsically scalable over wide surface areas. The magnetic properties of the resulting screens are studied experimentally at 77 K and numerically. Both bulk samples and coated conductors (HTS tapes) are potential candidates for building a scalable HTS screen. Bulks are able to efficiently attenuate the flux density behind the screen, but growing bulks whose surface area exceeds a few tens of cm² remains a challenge. By contrast, HTS tapes can be arranged into jointless closed loops, thereby forming naturally scalable HTS structures; however, the opposite flux density that they generate, and thus their screening effect, is very limited in their central region. The first key achievement of this thesis is therefore the introduction, investigation, optimisation, and physical interpretation of a hybrid HTS screen topology that combines a disc-shaped bulk with coaxially arranged closed HTS loops. This hybrid screen topology is validated experimentally by studying a 30 mm-diameter disc-shaped GdBCO bulk surrounded by closed-loop coated conductors, first subjected to a moderate (≤ 100 mT) inhomogeneous magnetic field and at small scale (typical dimension of the source coil ≤ 100 mm). With four concentric loops, the screened surface area reaches a record value of ∼72 cm², which is around 9 times larger than with the bulk alone. Importantly, the remarkable improvement brought by this hybrid screen can be explained physically: it is due to the bulk diverting the flux lines towards a region where the closed loops can oppose the best to them, i.e. at their vicinity. By exploiting measurements of a bespoke 3-axis cryogenic Hall probe, the screening factor SF (defined as the position-wise ratio between the applied field and the flux density with the screen) can be significantly enhanced by arranging the loops to minimise the transverse component of the induced field. Numerically, the influence of the physical and geometric parameters of the screens is studied using a 2D axisymmetric finite element model, therefore establishing general design rules. The main outcomes can be summarised as follows: (i) high-Jc materials are mandatory to maintain the screening properties up to high fields, whereas the maximum SF and screened surface area at low field are primarily limited by the geometry of the screen, (ii) the spacing between successive loops should be sufficiently small, with an optimal spacing that can be determined and physically explained by analysing the different paths followed by the flux lines, and (iii) the screening performance depends on the inhomogeneity of the applied field, with strongly inhomogeneous fields being attenuated more effectively as the natural curvature of the flux lines is beneficial for the screening effect. The evolution of SF when several field excitation cycles are applied, as it is the case in practical applications, is also important to consider. Significantly, more than 95% of the screening factor obtained after the first ramp remain after 300 cycles, which is explained physically based on the current density distribution in the HTS screen. The slight decay of SF after several cycles is shown to be smaller than the decay that would result from conventional current relaxation if the magnetic field is kept constant after the first increasing ramp. Another key achievement of this thesis is to demonstrate at larger scale the potential of an HTS screen naturally scalable in one direction. It consists of four 60 mm-diameter disc-shaped bulks arranged side by side with interleaved closed loops and short stacked tapes. The experiments are carried out under an inhomogeneous magnetic field (up to ~150 mT) produced by a large HTS coil (typical dimension > 200 mm) which served as a demonstrator for an HTS wind turbine rotor, available at the Technical University of Denmark (DTU). The addition of HTS tapes effectively mitigates the presence of gaps between the bulks, providing a lightweight and efficient solution to extend the screened surface area. At low field, a suitable screening factor (SF ≥ 1.5) is achieved over at least 138 mm along the scalable direction. The role of the different parts of the screen is confirmed with trapped field measurements. The screening results are analysed and interpreted by separately considering the axial and transverse magnetic field components, as well as the diversion of the flux lines. The influence of the inhomogeneity of the applied field is also examined.