Cancer is one of the leading causes of death worldwide. Conventional cancer treatment relies on radiotherapy and chemotherapy, but both methods bring severe side effects to patients, as these therapies not only attack cancer cells but also damage normal cells. Anticancer peptides (ACPs) are a promising alternative as therapeutic agents that are efficient and selective against tumor cells. Here, we propose a deep learning method based on convolutional neural networks to predict biological activity (EC50, LC50, IC50, and LD50) against six tumor cells, including breast, colon, cervix, lung, skin, and prostate. We show that models derived with multitask learning achieve better performance than conventional single-task models. In repeated 5-fold cross validation using the CancerPPD data set, the best models with the applicability domain defined obtain an average mean squared error of 0.1758, Pearson's correlation coefficient of 0.8086, and Kendall's correlation coefficient of 0.6156. As a step toward model interpretability, we infer the contribution of each residue in the sequence to the predicted activity by means of feature importance weights derived from the convolutional layers of the model. The present method, referred to as xDeep-AcPEP, will help to identify effective ACPs in rational peptide design for therapeutic purposes. The data, script files for reproducing the experiments, and the final prediction models can be downloaded from http://github.com/chen709847237/xDeep-AcPEP. The web server to directly access this prediction method is at https://app.cbbio.online/acpep/home.