Thanks to the ability of providing an immersive and interactive experience, the uptake of 360 degree image content has been rapidly growing in consumer and industrial applications. Compared to planar 2D images, saliency prediction for 360 degree images is more challenging due to their high resolutions and spherical viewing ranges. Currently, most high-performance saliency prediction models for omnidirectional images (ODIs) rely on deeper or broader convolutional neural networks (CNNs), which benefit from CNNs' superior feature representation capabilities while suffering from their high computational costs. In this paper, inspired by the human visual cognitive process, i.e., human being's perception of a visual scene is always accomplished by multiple stages of analysis, we propose a novel multi-stage recurrent generative adversarial networks for ODIs dubbed MRGAN360, to predict the saliency maps stage by stage. At each stage, the prediction model takes as input the original image and the output of the previous stage and outputs a more accurate saliency map. We employ a recurrent neural network among adjacent prediction stages to model their correlations, and exploit a discriminator at the end of each stage to supervise the output saliency map. In addition, we share the weights among all the stages to obtain a lightweight architecture that is computationally cheap. Extensive experiments are conducted to demonstrate that our proposed model outperforms the state-of-the-art model in terms of both prediction accuracy and model size.