Vessel segmentation in fundus is a key diagnostic capability in ophthalmology, and there are various challenges remained in this essential task. Early approaches indicate that it is often difficult to obtain desirable segmentation performance on thin vessels and boundary areas due to the imbalance of vessel pixels with different thickness levels. In this paper, we propose a novel two-stream Meticulous-Processing Network (MP-Net) for tackling this problem. To pay more attention to the thin vessels and boundary areas, we firstly propose an efficient hierarchical model automatically stratifies the ground-truth masks into different thickness levels. Then a novel two-stream adversarial network is introduced to use the stratification results with a balanced loss function and an integration operation to achieve a better performance, especially in thin vessels and boundary areas detecting. Our model is proved to outperform state-of-the-art methods on DRIVE, STARE, and CHASE_DB1 datasets.