Technological advances have enabled low-input RNA-sequencing, paving the way for assaying transcriptome variation in spatial contexts, including in tissues. While the generation of spatially resolved transcriptome maps is increasingly feasible, computational methods for analysing the resulting data are not established. Existing analysis strategies either ignore the spatial component of gene expression variation, or require discretization of the cells into coarse grained groups. To address this, we have developed SpatialDE, a computational framework for identifying and characterizing spatially variable genes. Our method generalizes variable gene selection, as used in population-and single-cell studies, to spatial expression profiles. To illustrate the broad utility of our approach, we apply SpatialDE to spatial transcriptomics data, and to data from single cell methods based on multiplexed in situ hybridisation (SeqFISH and MERFISH). SpatialDE enables the statistically robust identification of spatially variable genes, thereby identifying genes with known disease implications, several of which are missed by conventional variable gene selection. Additionally, to enable gene-expressed based histology, SpatialDE implements a spatial gene clustering model which we call “automatic expression histology,” allowing to classify genes into groups with distinct spatial patterns.