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Lensless imaging has emerged as a potential solution towards realizing ultra-miniature cameras by eschewing the bulky lens in a traditional camera. Without a focusing lens, the lensless cameras rely on computational algorithms to recover the scenes from multiplexed measurements. However, the current iterative-optimization-based reconstruction algorithms produce noisier and perceptually poorer images. In this work, we propose a non-iterative deep learning-based reconstruction approach that results in orders of magnitude improvement in image quality for lensless reconstructions. Our approach, called FlatNet, lays down a framework for reconstructing high-quality photorealistic images from mask-based lensless cameras, where the camera's forward model formulation is known. FlatNet consists of two stages: (1) an inversion stage that maps the measurement into a space of intermediate reconstruction by learning parameters within the forward model formulation, and (2) a perceptual enhancement stage that improves the perceptual quality of this intermediate reconstruction. These stages are trained together in an end-to-end manner. We show high-quality reconstructions by performing extensive experiments on real and challenging scenes using two different types of lensless prototypes: one which uses a separable forward model and another, which uses a more general non-separable cropped-convolution model. Our end-to-end approach is fast, produces photorealistic reconstructions, and is easy to adopt for other mask-based lensless cameras.