The aye-aye ( Daubentonia madagascariensis ) is one of the 25 most endangered primate species in the world, maintaining amongst the lowest genetic diversity of any primate measured to date. Characterizing patterns of genetic variation within aye-aye populations, and the relative influences of neutral and selective processes in shaping that variation, is thus important for future conservation efforts. In this study, we performed the first whole-genome scans for recent positive and balancing selection in the species, utilizing high-coverage population genomic data from newly sequenced individuals. We generated null thresholds for our genomic scans by creating an evolutionarily appropriate baseline model that incorporates the demographic history of this aye-aye population, and identified a small number of candidate genes. Most notably, a suite of genes involved in olfaction - a key trait in these nocturnal primates - were identified as experiencing long-term balancing selection. We also conducted analyses to quantify the expected statistical power to detect positive and balancing selection in this population using site frequency spectrum-based inference methods, once accounting for the potentially confounding contributions of population history, recombination and mutation rate variation, and purifying and background selection. This work, presenting the first high-quality, genome-wide polymorphism data across the functional regions of the aye-aye genome, thus provides important insights into the landscape of episodic selective forces in this highly endangered species.