Positron emission tomography (PET) is a cornerstone of modern radiology. The ability to detect cancer and metastases in whole body scans fundamentally changed cancer diagnosis and treatment. One of the main bottlenecks in the clinical application is the time it takes to reconstruct the anatomical image from the deluge of data in PET imaging. State-of-the art methods based on expectation maximization can take hours for a single patient and depend on manual fine-tuning. This results not only in financial burden for hospitals but more importantly leads to less efficient patient handling, evaluation, and ultimately diagnosis and treatment for patients. To overcome this problem we present a novel PET image reconstruction technique based on a deep convolutional encoder-decoder network, that takes PET sinogram data as input and directly outputs full PET images. Using realistic simulated data, we demonstrate that our network is able to reconstruct images >100 times faster, and with comparable image quality (in terms of root mean squared error) relative to conventional iterative reconstruction techniques.