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Chronic Venous Disorders (CVD) of the lower limbs are one of the most prevalent medical conditions, affecting 35% of adults in Europe and North America. Due to the exponential growth of the aging population and the worsening of CVD with age, it is expected that the healthcare costs and the resources needed for the treatment of CVD will increase in the coming years. The early diagnosis of CVD is fundamental in treatment planning, while the monitoring of its treatment is fundamental to assess a patient's condition and quantify the evolution of CVD. However, correct diagnosis relies on a qualitative approach through visual recognition of the various venous disorders, being time-consuming and highly dependent on the physician's expertise. In this paper, we propose a novel automatic strategy for the joint segmentation and classification of CVDs. The strategy relies on a multi-task deep learning network, denominated VENet, that simultaneously solves segmentation and classification tasks, exploiting the information of both tasks to increase learning efficiency, ultimately improving their performance. The proposed method was compared against state-of-the-art strategies in a dataset of 1376 CVD images. Experiments showed that the VENet achieved a classification performance of 96.4%, 96.4%, and 97.2% for accuracy, precision, and recall, respectively, and a segmentation performance of 75.4%, 76.7.0%, 76.7% for the Dice coefficient, precision, and recall, respectively. The joint formulation increased the robustness of both tasks when compared to the conventional classification or segmentation strategies, proving its added value, mainly for the segmentation of small lesions. [ABSTRACT FROM AUTHOR]