Magnetically induced current densities, nuclear magnetic shieldings, and electric polarizabilities of planar ring-shaped hydrocarbons have been studied at the density-functional theory level using the Becke-Perdew (BP86) functional. The current densities were calculated using the Gauge-Including Magnetically Induced Current (GIMIC) method employing gauge-including atomic orbitals. The GIMIC calculations yield rules to estimate the global and local ring-current strengths as well as the current pathways for the hydrocarbon nanorings. For the overall antiaromatic molecules, aromatic groups such as benzene, naphthalene, anthracene, and pyrene moieties localize the ring current making the global ring currents vanish. The ability of the edge groups to localize the currents is related to the aromatic character of the molecule as a whole. The local ring current prefers to follow the edges of the group. Phenalenyl corner moieties are found to introduce strong global ring currents, whereas with fused benzene and pyrene corner groups the global ring current vanishes. Fused benzene rings in the corner or along the edge of overall antiaromatic molecules sustain local ring currents of about the same size as for a free benzene molecule. For the overall aromatic molecules, the global ring current is split along the bonds of the edge moieties, but the detailed division fulfilling Kirchhoff's current law is not easily predictable and must be calculated for each individual bond. At the phenalenyl corner moieties, the global ring current follows the innermost route isolating the rest of the group from the main delocalization pathway. A hydrocarbon nanoring sustaining strong ring currents should be large and formally aromatic with many and large aromatic moieties along the edges. A clear correlation between the strength of the global ring currents and the size of the electric polarizabilities is obtained. The calculated H-1 NMR shieldings of a proton in immediate contact to the global ring current vary between 22 ppm and 67 ppm in the studied molecules. The trend correlates well with the global ring-current strengths, which are in the range of 0-88 nA/T. The C-13 NMR shieldings are also sensitive to the strength of the global ring current, but they vary less systematically and are not as good an indicator of the current strength as the hydrogen shieldings.