Transformer-based language models, such as BERT and its variants, have achieved state-of-the-art performance in several downstream natural language processing (NLP) tasks on generic benchmark datasets (e.g., GLUE, SQUAD, RACE). However, these models have mostly been applied to the resource-rich English language. In this paper, we present GREEK-BERT, a monolingual BERT-based language model for modern Greek. We evaluate its performance in three NLP tasks, i.e., part-of-speech tagging, named entity recognition, and natural language inference, obtaining state-of-the-art performance. Interestingly, in two of the benchmarks GREEK-BERT outperforms two multilingual Transformer-based models (M-BERT, XLM-R), as well as shallower neural baselines operating on pretrained word embeddings, by a large margin (5%-10%). Most importantly, we make both GREEK-BERT and our training code publicly available, along with code illustrating how GREEK-BERT can be fine-tuned for downstream NLP tasks. We expect these resources to boost NLP research and applications for modern Greek.