This paper proposes Personalized Diversity-promoting GAN (PD-GAN), a novel recommendation model to generate diverse, yet relevant recommendations. Specifically, for each user, a generator recommends a set of diverse and relevant items by sequentially sampling from a personalized Determinantal Point Process (DPP) kernel matrix. This kernel matrix is constructed by two learnable components: the general co-occurrence of diverse items and the user's personal preference to items. To learn the first component, we propose a novel pairwise learning paradigm using training pairs, and each training pair consists of a set of diverse items and a set of similar items randomly sampled from the observed data of all users. The second component is learnt through adversarial training against a discriminator which strives to distinguish between recommended items and the ground-truth sets randomly sampled from the observed data of the target user. Experimental results show that PD-GAN is superior to generate recommendations that are both diverse and relevant.