Subject of Research. The present work considers the problem of detecting replay attacks on voice biometric systems. Due to their simplicity, these attacks are more likely to be used by the imposters, and that is why they are of special risk. This work describes the system for detecting replay attacks that was presented on the Automatic Speaker Verification Spoofing and Countermeasures (ASVspoof) Challenge 2017 focused on this problem.Method. We study the efficiency of deep learning approach in the described task, in particular, convolutional neural networks with Max-Feature-Map activation function. Main Results. Experimental results obtained on the Challenge corpora have demonstrated high performance of such approach in contrast to current state-of-the-art baseline systems. Our primary system achieved 6.73% EER on the evaluation part of the corpora which is 72% relative improvement over the ASVspoof 2017 baseline system. Practical Relevance. The results of the work can be applied in the field of voice biometrics. The presented methods can be used in systems of automatic speaker verification and identification for detecting spoofing attacks on them.