Old age is associated with a decline in cognitive function and an increase in neurodegenerative disease risk1. Brain ageing is complex and is accompanied by many cellular changes2. Furthermore, the influence that aged cells have on neighbouring cells and how this contributes to tissue decline is unknown. More generally, the tools to systematically address this question in ageing tissues have not yet been developed. Here we generate a spatially resolved single-cell transcriptomics brain atlas of 4.2 million cells from 20 distinct ages across the adult lifespan and across two rejuvenating interventions-exercise and partial reprogramming. We build spatial ageing clocks, machine learning models trained on this spatial transcriptomics atlas, to identify spatial and cell-type-specific transcriptomic fingerprints of ageing, rejuvenation and disease, including for rare cell types. Using spatial ageing clocks and deep learning, we find that T cells, which increasingly infiltrate the brain with age, have a marked pro-ageing proximity effect on neighbouring cells. Surprisingly, neural stem cells have a strong pro-rejuvenating proximity effect on neighbouring cells. We also identify potential mediators of the pro-ageing effect of T cells and the pro-rejuvenating effect of neural stem cells on their neighbours. These results suggest that rare cell types can have a potent influence on their neighbours and could be targeted to counter tissue ageing. Spatial ageing clocks represent a useful tool for studying cell-cell interactions in spatial contexts and should allow scalable assessment of the efficacy of interventions for ageing and disease.