Context. Magnetic fields play an important role in the formation and stabilization of spiral structures in galaxies, but the interaction between interstellar gas and magnetic fields has not yet been understood. In particular, the phenomenon of "magnetic arms" located between material arms is a mystery. Aims. The strength and structure of interstellar magnetic fields and their relation to spiral arms in gas and dust are investigated in the nearby and almost face-on spiral galaxy IC 342. Methods. The total and polarized radio continuum emission of IC 342 was observed with high spatial resolution in four wavelength bands with the Effelsberg and VLA telescopes. At λ6.2 cm the data from both telescopes were combined. I separated thermal and nonthermal (synchrotron) emission components with the help of the spectral index distribution and derived maps of the magnetic field strength, degree of magnetic field order, magnetic pitch angle, Faraday rotation measure, and Faraday depolarization. Results. IC 342 hosts a diffuse radio disk with an intensity that decreases exponentially with increasing radius. The frequency dependence of the scalelength of synchrotron emission indicates energy-dependent propagation of the cosmic-ray electrons, probably via the streaming instability. The equipartition strength of the total field in the main spiral arms is typically 15 μG, that of the ordered field about 5 μG. The total radio emission, observed with the VLA's high resolution, closely follows the dust emission in the infrared at 8 μm (Spitzer telescope) and 22 μm (WISE telescope). The polarized emission is not diffuse, but concentrated in spiral arms of various types: (1) a narrow arm of about 300 pc width, displaced inwards with respect to the eastern arm by about 200 pc, indicating magnetic fields compressed by a density wave; (2) a broad arm of 300-500 pc width around the northern arm with systematic variations in polarized emission, polarization angles, and Faraday rotation measures on a scale of about 2 kpc, indicative of a helically twisted flux tube generated by the Parker instability; (3) a rudimentary magnetic arm in an interarm region in the north-west; (4) several broad spiral arms in the outer galaxy, related to spiral arms in the total neutral gas; (5) short features in the outer south-western galaxy, probably distorted by tidal interaction. Faraday rotation of the polarization angles reveals an underlying regular field of only 0.5 μG strength with a large-scale axisymmetric spiral pattern, probably a signature of a mean-field α − Ω dynamo, and an about 10× stronger field that fluctuates on scales of a few 100 pc. The magnetic field around the bar in the central region of IC 342 resembles that of large barred galaxies; it has a regular spiral pattern with a large pitch angle, is directed outwards, and is opposite to the large-scale regular field in the disk. Polarized emission at λ20.1 cm is strongly affected by Faraday depolarization in the western and northern parts of the galaxy. Helical fields extending fro...