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An intense muon beam is mandatory for next-generation experiments to search for lepton flavor violating processes in the muon sector. The COMET experiment, J-PARC E21, aims to search for muon to electron conversion with an unprecedented sensitivity. All the components of the experiment, such as the pion production target, the muon stopping target and the tracker are embedded in superconducting solenoids, resulting in a total length longer than 30 m. The pions are captured in a 5 T solenoid magnet with a diameter of 1.3 m, and decay to muons in the subsequent 3 T toroidal magnets over a length of 10 m. The pion capture solenoid is designed to be as small as possible, however, thick shielding is necessary within the solenoid to avoid severe radiation from the target. Aluminum-stabilized NbTi superconducting wire is employed to reduce the cold mass and energy deposited in it. The damage of the conductor of the coils should be estimated carefully, since the expected neutron fluence reaches 10^22\ neutrons/m^2. This paper describes the design of the solenoid magnets and R&D programs. [ABSTRACT FROM AUTHOR]