Dementia is one of the most common neurological syndromes in the world. Usually, diagnoses are made based on paper-and-pencil tests and scored depending on personal judgments of experts. This technique can introduce errors and has high inter-rater variability. To overcome these issues, we present an automatic assessment of the widely used paper-based clock-drawing test by means of deep neural networks. Our study includes a comparison of three modern architectures: VGG16, ResNet-152, and DenseNet-121. The dataset consisted of 1315 individuals. To deal with the limited amount of data, which also included several dementia types, we used optimization strategies for training the neural network. The outcome of our work is a standardized and digital estimation of the dementia screening result and severity level for an individual. We achieved accuracies of 96.65% for screening and up to 98.54% for scoring, overcoming the reported state-of-the-art as well as human accuracies. Due to the digital format, the paper-based test can be simply scanned by using a mobile device and then be evaluated also in areas where there is a staff shortage or where no clinical experts are available.