Liver vessel segmentation is fast becoming a key instrument in the diagnosis and surgical planning of liver diseases. In clinical practice, liver vessels are normally manual annotated by clinicians on each slice of CT images, which is extremely laborious. Several deep learning methods existed for liver vessel segmentation, however, promoting the performance of segmentation remains a major challenge due to the large variations and complex structure of liver vessels. Previous methods mainly using existing UNet architecture, but not all features of the encoder are useful for segmentation and some even cause interferences. To overcome this problem, we propose a novel deep neural network for liver vessel segmentation, called LVSNet, which employed special designs to obtain the accurate structure of the liver vessel. Specifically, we design Attention-Guided Concatenation (AGC) module to adaptively select the useful context features from low-level features guided by high-level features. The proposed AGC module focuses on capturing rich complemented information to obtain more details. In addition, we introduce an innovative multi-scale fusion block by constructing hierarchical residual-like connections within one single residual block, which is great importance for effectively linking the local blood vessel fragments together. Furthermore, we construct a new dataset containing 40 thin thickness cases (0.625mm) which consist of CT volumes and annotated vessels. To evaluate the effectiveness of the method with minor vessel, we also propose an automatic stratification method to split major and minor liver vessels. Extensive experimental results demonstrate that the proposed LVSNet outperforms previous methods on liver vessel segmentation datasets. Additionally, we conduct a series of ablation studies that comprehensively support the superiority of the underlying concepts.