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Context: Identifying potential vulnerable code is important to improve the security of our software systems. However, the manual detection of software vulnerabilities requires expert knowledge and is time-consuming, and must be supported by automated techniques. Objective: Such automated vulnerability detection techniques should achieve a high accuracy, point developers directly to the vulnerable code fragments, scale to real-world software, generalize across the boundaries of a specific software project, and require no or only moderate setup or configuration effort. Method: In this article, we present VUDENC (Vulnerability Detection with Deep Learning on a Natural Codebase), a deep learning-based vulnerability detection tool that automatically learns features of vulnerable code from a large and real-world Python codebase. VUDENC applies a word2vec model to identify semantically similar code tokens and to provide a vector representation. A network of long-short-term memory cells (LSTM) is then used to classify vulnerable code token sequences at a fine-grained level, highlight the specific areas in the source code that are likely to contain vulnerabilities, and provide confidence levels for its predictions. Results: To evaluate VUDENC, we used 1,009 vulnerability-fixing commits from different GitHub repositories that contain seven different types of vulnerabilities (SQL injection, XSS, Command injection, XSRF, Remote code execution, Path disclosure, Open redirect) for training. In the experimental evaluation, VUDENC achieves a recall of 78%-87%, a precision of 82%-96%, and an F1 score of 80%-90%. VUDENC's code, the datasets for the vulnerabilities, and the Python corpus for the word2vec model are available for reproduction. Conclusions: Our experimental results suggest...