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Face aging is to render a given face to predict its future appearance, which plays an important role in the information forensics and security field as the appearance of the face typically varies with age. Although impressive results have been achieved with conditional generative adversarial networks (cGANs), the existing cGANs-based methods typically use a single network to learn various aging effects between any two different age groups. However, they cannot simultaneously meet three essential requirements of face aging -- including image quality, aging accuracy, and identity preservation -- and usually generate aged faces with strong ghost artifacts when the age gap becomes large. Inspired by the fact that faces gradually age over time, this paper proposes a novel progressive face aging framework based on generative adversarial network (PFA-GAN) to mitigate these issues. Unlike the existing cGANs-based methods, the proposed framework contains several sub-networks to mimic the face aging process from young to old, each of which only learns some specific aging effects between two adjacent age groups. The proposed framework can be trained in an end-to-end manner to eliminate accumulative artifacts and blurriness. Moreover, this paper introduces an age estimation loss to take into account the age distribution for an improved aging accuracy, and proposes to use the Pearson correlation coefficient as an evaluation metric measuring the aging smoothness for face aging methods. Extensively experimental results demonstrate superior performance over existing (c)GANs-based methods, including the state-of-the-art one, on two benchmarked datasets. The source code is available at~\url{https://github.com/Hzzone/PFA-GAN}.