  The KATRIN experiment is a next-generation direct neutrino mass experiment
with a sensitivity of 0.2 eV (90% C.L.) to the effective mass of the electron
neutrino. It measures the tritium $\beta$-decay spectrum close to its endpoint
with a spectrometer based on the MAC-E filter technique. The $\beta$-decay
electrons are guided by a magnetic field that operates in the mT range in the
central spectrometer volume; it is fine-tuned by a large-volume air coil system
surrounding the spectrometer vessel. The purpose of the system is to provide
optimal transmission properties for signal electrons and to achieve efficient
magnetic shielding against background. In this paper we describe the technical
design of the air coil system, including its mechanical and electrical
properties. We outline the importance of its versatile operation modes in
background investigation and suppression techniques. We compare magnetic field
measurements in the inner spectrometer volume during system commissioning with
corresponding simulations, which allows to verify the system's functionality in
fine-tuning the magnetic field configuration. This is of major importance for a
successful neutrino mass measurement at KATRIN.
