Combining multiple types of genomic, transcriptional, proteomic, and epigenetic datasets has the potential to reveal biological mechanisms across multiple scales, and may lead to more accurate models for clinical decision support. Developing efficient models that can derive clinical outcomes from high-dimensional data remains problematical; challenges include the integration of multiple types of omics data, inclusion of biological background knowledge, and developing machine learning models that are able to deal with this high dimensionality while having only few samples from which to derive a model. We developed DeepMOCCA, a framework for multi-omics cancer analysis. We combine different types of omics data using biological relations between genes, transcripts, and proteins, combine the multi-omics data with background knowledge in the form of protein-protein interaction networks, and use graph convolution neural networks to exploit this combination of multi-omics data and background knowledge. DeepMOCCA predicts survival time for individual patient samples for 33 cancer types and outperforms most existing survival prediction methods. Moreover, DeepMOCCA includes a graph attention mechanism which prioritizes driver genes and prognostic markers in a patient-specific manner; the attention mechanism can be used to identify drivers and prognostic markers within cohorts and individual patients.