Deep Learning (DL) systems are key enablers for engineering intel- ligent applications due to their ability to solve complex tasks such as image recognition and machine translation. Nevertheless, using DL systems in safety- and security-critical applications requires to provide testing evidence for their dependable operation. Recent research in this direction focuses on adapting testing criteria from traditional software engineering as a means of increasing confi- dence for their correct behaviour. However, they are inadequate in capturing the intrinsic properties exhibited by these systems. We bridge this gap by introducing DeepImportance, a systematic testing methodology accompanied by an Importance-Driven (IDC) test adequacy criterion for DL systems. Applying IDC enables to establish a layer-wise functional understanding of the importance of DL system components and use this information to guide the generation of semantically-diverse test sets. Our empirical evalua- tion on several DL systems, across multiple DL datasets and with state-of-the-art adversarial generation techniques demonstrates the usefulness and effectiveness of DeepImportance and its ability to guide the engineering of more robust DL systems.