Recent progress on fine-grained visual recognition and visual question answering has featured Bilinear Pooling, which effectively models the 2$^{nd}$nd order interactions across multi-modal inputs. Nevertheless, there has not been evidence in support of building such interactions concurrently with attention mechanism for image captioning. In this paper, we introduce a unified attention block -- X-Linear attention block, that fully employs bilinear pooling to selectively capitalize on visual information or perform multi-modal reasoning. Technically, X-Linear attention block simultaneously exploits both the spatial and channel-wise bilinear attention distributions to capture the 2$^{nd}$nd order interactions between the input single-modal or multi-modal features. Higher and even infinity order feature interactions are readily modeled through stacking multiple X-Linear attention blocks and equipping the block with Exponential Linear Unit (ELU) in a parameter-free fashion, respectively. Furthermore, we present X-Linear Attention Networks (dubbed as X-LAN) that novelly integrates X-Linear attention block(s) into image encoder and sentence decoder of image captioning model to leverage higher order intra- and inter-modal interactions. The experiments on COCO benchmark demonstrate that our X-LAN obtains to-date the best published CIDEr performance of 132.0% on COCO Karpathy test split. When further endowing Transformer with X-Linear attention blocks, CIDEr is boosted up to 132.8%. Source code is available at \url{https://github.com/Panda-Peter/image-captioning}.