Recent advances in single cell omics technologies enable the individual or joint profiling of cellular measurements including gene expression, epigenetic features, chromatin structure and DNA sequences. Currently, most single-cell analysis pipelines are cluster-centric, i.e., they first cluster cells into non-overlapping cellular states and then extract their defining genomic features. These approaches assume that discrete clusters correspond to biologically relevant subpopulations and do not explicitly model the interactions between different feature types. However, cellular processes are defined in individual cells and inherently involve multiple genomic features that interact with each other and together provide complementary views on principles of gene regulation. In addition, single-cell methods are generally designed for a particular task as distinct single-cell problems are formulated differently. To address these current shortcomings, we present SIMBA, a single-cell embedding method that embeds single cells along with their defining features, such as genes, chromatin accessible regions, and transcription factor binding sequences, into a common latent space. By leveraging the co-embedding of cells and features, SIMBA allows for cellular heterogeneity study, clustering-free marker discovery, gene regulation inference, batch effect removal, and omics data integration. SIMBA has been extensively applied to scRNA-seq, scATAC-seq, and dual-omics data. We show that SIMBA provides a single framework that allows diverse single-cell analysis problems to be formulated in a common way and thus simplifies the development of new analyses and integration of other single-cell modalities.