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Malicious software (malware) classification offers a unique challenge for continual learning (CL) regimes due to the volume of new samples received on a daily basis and the evolution of malware to exploit new vulnerabilities. On a typical day, antivirus vendors receive hundreds of thousands of unique pieces of software, both malicious and benign, and over the course of the lifetime of a malware classifier, more than a billion samples can easily accumulate. Given the scale of the problem, sequential training using continual learning techniques could provide substantial benefits in reducing training and storage overhead. To date, however, there has been no exploration of CL applied to malware classification tasks. In this paper, we study 11 CL techniques applied to three malware tasks covering common incremental learning scenarios, including task, class, and domain incremental learning (IL). Specifically, using two realistic, large-scale malware datasets, we evaluate the performance of the CL methods on both binary malware classification (Domain-IL) and multi-class malware family classification (Task-IL and Class-IL) tasks. To our surprise, continual learning methods significantly underperformed naive Joint replay of the training data in nearly all settings – in some cases reducing accuracy by more than 70 percentage points. A simple approach of selectively replaying 20% of the stored data achieves better performance, with 50% of the training time compared to Joint replay. Finally, we discuss potential reasons for the unexpectedly poor performance of the CL techniques, with the hope that it spurs further research on developing techniques that are more effective in the malware classification domain.