We investigate the general problem of reproducing color images on an offset press using custom inks in any combination and number. While this problem has been explored previously for the case of two inks, there are a number of new mathematical and algorithmic challenges that arise as the number of inks increases. These challenges include more complex gamut mapping strategies, more efficient ink selection strategies, and fast and numerically accurate methods for computing ink separations in situations that may be either over- or under-constrained. In addition, the demands of high-quality color printing require an accurate physical model of the colors that result from overprinting multiple inks using halftoning, including the effects of trapping, dot gain, and the interreflection of light between ink layers. In this paper, we explore these issues related to printing with multiple custom inks, and address them with new algorithms and physical models. Finally, we present some printed examples demonstrating the promise of our methods.