Anomaly detection in video is a challenging computer vision problem. Due to the lack of anomalous events at training time, anomaly detection requires the design of learning methods without full supervision. In this paper, we approach anomalous event detection in video through self-supervised and multi-task learning at the object level. We first utilize a pre-trained detector to detect objects. Then, we train a 3D convolutional neural network to produce discriminative anomaly-specific information by jointly learning multiple proxy tasks: three self-supervised and one based on knowledge distillation. The self-supervised tasks are: (i) discrimination of forward/backward moving objects (arrow of time), (ii) discrimination of objects in consecutive/intermittent frames (motion irregularity) and (iii) reconstruction of object-specific appearance information. The knowledge distillation task takes into account both classification and detection information, generating large prediction discrepancies between teacher and student models when anomalies occur. To the best of our knowledge, we are the first to approach anomalous event detection in video as a multi-task learning problem, integrating multiple self-supervised and knowledge distillation proxy tasks in a single architecture. Our lightweight architecture outperforms the state-of-the-art methods on three benchmarks: Avenue, ShanghaiTech and UCSD Ped2. Additionally, we perform an ablation study demonstrating the importance of integrating self-supervised learning and normality-specific distillation in a multi-task learning setting.