Most populations are spread over spatial ranges much bigger than any one individual will explore in its lifetime. How does the simple fact of this spatial structure affect adaptive evolution and genetic diversity? We quantify how space interacts with recombination (sex) to slow down or speed up adaptation by changing the amount of clonal interference among beneficial mutations, and to change how adaptation reshapes neutral diversity through genetic hitchhiking. We also show how the spatial and temporal variation in genetic diversity revealed by sequencing can be used as a lens to watch how organisms move, capturing rare jumps that have large effects on populations' evolution.