Epigenomics is the study of the physical modifications, associations and conformations of genomic DNA sequences, with the aim of linking these with epigenetic memory, cellular identity and tissue-specific functions. While current techniques in the field are characterizing the average epigenomic features across large cell ensembles, the increasing interest in the epigenetics within complex and heterogeneous tissues is driving the development of single-cell epigenomics. We review emerging single-cell methods for capturing DNA methylation, chromatin accessibility, histone modifications, chromosome conformation and replication dynamics. Together, these techniques are rapidly becoming a powerful tool in studies of cellular plasticity and diversity, as seen in stem cells and cancer.