Graded-index optical fiber probes suitable for focusing or collimating the output of an optical fiber at a wavelength of 1.3 µm have become an enabling technology in optical coherence tomography imaging applications for in vivo bioimaging. Such fiber tips however remain uncommon in other photonics applications. This paper provides the first numerical study of graded-index fiber tips covering a broad range of wavelengths spanning from the UV to short-infrared. The wavelength dependency and the influence of probe geometry on performance characteristics such as far-field divergence angle, spot size and working distance are analyzed. The paper yields easily accessible design guidelines for the fabrication of collimating or focusing fiber tips. Fiber collimators have considerable potential for use in free-space systems and could benefit a range of devices such as variable attenuators, dynamic wavelength equalisers and large 3D optical cross-connect switches, whereas focusing fiber tips have applications in high-resolution imaging.