Cell tracking is an omnipresent image analysis task in live-cell microscopy. It is similar to multiple object tracking (MOT), however, each frame contains hundreds of similar-looking objects that can divide, making it a challenging problem. Current state-of-the-art approaches follow the tracking-by-detection paradigm, i.e. first all cells are detected per frame and successively linked in a second step to form biologically consistent cell tracks. Linking is commonly solved via discrete optimisation methods, which require manual tuning of hyperparameters for each dataset and are therefore cumbersome to use in practice. Here we propose Trackastra, a general purpose cell tracking approach that uses a simple transformer architecture to directly learn pairwise associations of cells within a temporal window from annotated data. Importantly, unlike existing transformer-based MOT pipelines, our learning architecture also accounts for dividing objects such as cells and allows for accurate tracking even with simple greedy linking, thus removing the requirement for a complex linking step. The proposed architecture operates on the full spatio-temporal context of detections within a time window by avoiding the computational burden of processing dense images. We show that our tracking approach performs on par with or better than highly tuned state-of-the-art cell tracking algorithms for various biological datasets, such as bacteria, cell cultures and fluorescent particles. We provide code at https://github.com/*.