The recent development of a CRISPR-Cas9-based homing system for the suppression of Anopheles gambiae is encouraging; however, with current designs, the slow emergence of homing-resistant alleles is expected to result in suppressed populations rapidly rebounding, as homing-resistant alleles have a significant fitness advantage over functional, population-suppressing homing alleles. To explore this concern, we develop a mathematical model to estimate tolerable rates of homing-resistant allele generation to suppress a wild population of a given size. Our results suggest that, to achieve meaningful population suppression, tolerable rates of resistance allele generation are orders of magnitude smaller than those observed for current designs for CRISPR-Cas9-based homing systems. To remedy this, we theoretically explore a homing system architecture in which guide RNAs (gRNAs) are multiplexed, increasing the effective homing rate and decreasing the effective resistant allele generation rate. Modeling results suggest that the size of the population that can be suppressed increases exponentially with the number of multiplexed gRNAs and that, with four multiplexed gRNAs, a mosquito species could potentially be suppressed on a continental scale. We also demonstrate successful proof-of-principle use of multiplexed ribozyme flanked gRNAs to induce mutations in vivo in Drosophila melanogaster - a strategy that could readily be adapted to engineer stable, homing-based drives in relevant organisms.