Stellar population models of absorption-line indices are an important tool for the analysis of stellar population spectra. They are most accurately modelled through empirical calibrations of absorption-line indices with the stellar parameters such as effective temperature, metallicity and surface gravity, which are the so-called fitting functions. Here we present new empirical fitting functions for the 25 optical Lick absorption-line indices based on the new stellar library Medium resolution INT Library of Empirical Spectra (MILES). The major improvements with respect to the Lick/IDS library are the better sampling of stellar parameter space, a generally higher signal-to-noise ratio and a careful flux calibration. In fact, we find that errors on individual index measurements in MILES are considerably smaller than in Lick/IDS. Instead, we find the rms of the residuals between the final fitting functions and the data to be dominated by errors in the stellar parameters. We provide fitting functions for both Lick/IDS and MILES spectral resolutions and compare our results with other fitting functions in the literature.