Deep Learning (DL) systems are key enablers for engineering intelligent applications. Nevertheless, using DL systems in safety- and security-critical applications requires to provide testing evidencefor their dependable operation. We introduce 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 assess the semantic diversityof a test set. Our empirical evaluation on several DL systems and across multiple DL datasets demonstrates the usefulness and effectiveness of DeepImportance.