Feshbach resonances induced by dc magnetic fields are a powerful tool to engineer inter- actions in ultracold quantum gases and they have led to significant advances in many-body physics, condensed matter physics, precision spectroscopy, quantum information processing, creating ultracold molecules, and facilitating evaporative cooling in optical dipole traps [1– 5]. A number of related techniques make use of laser light [6], dc electric fields [7], or a combination of magnetic and optical fields [8] to shift the atomic energy levels in and out of resonance with those of a transient collision complex, thereby inducing resonant variation of the scattering length [1]. Dressed state potentials induced by external radio frequency (rf) and microwave fields have remarkable properties. On atom chips [9], where substantial strength and gradients can be reached in the nearfield, complex potential landscapes (such as double-well and ring traps) can be created, which are ideal for matter wave interferometry [10–13] and exploring quantum many body physics [14]. Moreover, rf-dressed atoms are immune to spontaneous decay and it was suggested that low-frequency electromagnetic radiation can be used to control atomic collisions at ultralow temperatures [11]. Early studies of atomic collisions in the presence of low-frequency electromagnetic radia- tion date back to the development of microwave traps for neutral atoms [15], the studies of dipolar relaxation of spin-polarized hydrogen atoms in a microwave trap [16] and detailed analysis of rf-induced evaporative cooling [17]. In the last few years, the creation of weakly bound KRb and Cr$_{2}$ molecules by rf-assisted association of ultracold atoms was demon- strated [18, 19]. Very recently, multichannel quantum defect theory was applied to describe collisions of ultracold Rb atoms in rf fields [20]. In this Rapid Communication, we develop a rigorous quantum theory of atomic collisions in the presence of rf fields. Expanding on the work of Agosta et al. [16], we combine the coupled-channel expansion with the field-dressed formalism to describe the collision complex in an external rf field. This allows us to calculate the scattering length of ultracold Rb atoms as a function of the amplitude and frequency of the rf field. We demonstrate that collisions of ultracold atoms are sensitive to rf fields of moderate strength easily accessible in atom chip experiments. In particular, rf fields may be used to manipulate magnetic Feshbach resonances and to create new, rf-induced Feshbach resonances, whose positions and widths can be tuned by changing the rf field parameters. The Hamiltonian for two alkali-metal atoms A and B colliding in the presence of super- 2