Cross-spectral face recognition (CFR) refers to recognizing individuals using face images stemming from different spectral bands, such as infrared versus visible. While CFR is inherently more challenging than classical face recognition due to significant variation in facial appearance caused by the modality gap, it is useful in many scenarios including night-vision biometrics and detecting presentation attacks. Recent advances in deep neural networks (DNNs) have resulted in significant improvement in the performance of CFR systems. Given these developments, the contributions of this survey are three-fold. First, we provide an overview of CFR, by formalizing the CFR problem and presenting related applications. Secondly, we discuss the appropriate spectral bands for face recognition and discuss recent CFR methods, placing emphasis on deep neural networks. In particular we describe techniques that have been proposed to extract and compare heterogeneous features emerging from different spectral bands. We also discuss the datasets that have been used for evaluating CFR methods. Finally, we discuss the challenges and future lines of research on this topic.