Automatic liver tumor segmentation from 3D Computed Tomography (CT) is a necessary prerequisite in the interventions of hepatic abnormalities and surgery planning. However, accurate liver tumor segmentation remains challenging due to the large variability of tumor sizes and inhomogeneous texture. Recent advances based on Fully Convolutional Network (FCN) in liver tumor segmentation draw on success of learning discriminative multi-level features. In this paper, we propose a Deep Attentive Refinement Network (DARN) for improved liver tumor segmentation from CT volumes by fully exploiting both low and high level features embedded in different layers of FCN. Different from existing works, we exploit attention mechanism to leverage the relation of different levels of features encoded in different layers of FCN. Specifically, we introduce a Semantic Attention Refinement (SemRef) module to selectively emphasize global semantic information in low level features with the guidance of high level ones, and a Spatial Attention Refinement (SpaRef) module to adaptively enhance spatial details in high level features with the guidance of low level ones. We evaluate our network on the public MICCAI 2017 Liver Tumor Segmentation Challenge dataset (LiTS dataset) and it achieves state-of-the-art performance. The proposed refinement modules are an effective strategy to exploit multi-level features and has great potential to generalize to other medical image segmentation tasks.