Multimodal data is rapidly growing in many fields of science and engineering, including single-cell biology. We introduce MultiMAP, a novel algorithm for dimensionality reduction and integration. MultiMAP can integrate any number of datasets, leverages features not present in all datasets, is not restricted to a linear mapping, allows the user to specify the influence of each dataset, and is extremely scalable to large datasets. We apply MultiMAP to single-cell transcriptomics, chromatin accessibility, methylation, and spatial data and show that it outperforms current approaches. On a new thymus dataset, we use MultiMAP to integrate cells along a temporal trajectory. This enables quantitative comparison of transcription factor expression and binding site accessibility over the course of T cell differentiation, revealing patterns of expression versus binding site opening kinetics.