Protein sequence alignment is a key component of most bioinformatics pipelines to study the structures and functions of proteins. Aligning highly divergent sequences remains, however, a difficult task that current algorithms often fail to perform accurately, leaving many proteins or open reading frames poorly annotated. Here we leverage recent advances in deep learning for language modeling and differentiable programming to propose DEDAL (deep embedding and differentiable alignment), a flexible model to align protein sequences and detect homologs. DEDAL is a machine learning-based model that learns to align sequences by observing large datasets of raw protein sequences and of correct alignments. Once trained, we show that DEDAL improves by up to two- or threefold the alignment correctness over existing methods on remote homologs and better discriminates remote homologs from evolutionarily unrelated sequences, paving the way to improvements on many downstream tasks relying on sequence alignment in structural and functional genomics.