Coevolution--reciprocal evolutionary change in interacting species--is one of the central biological processes organizing the web of life, and most species are involved in one or more coevolved interactions. We have learned in recent years that coevolution is a highly dynamic process that continually reshapes interactions among species across ecosystems, creating geographic mosaics over timescales sometimes as short as thousands or even hundreds of years. If we take that as our starting point, what should we now be asking about the coevolutionary process? Here I suggest five major questions that we need to answer if we are to understand how coevolution shapes the web of life. How evolutionarily dynamic is specialization to other species, and what is the role of coevolutionary alternation in driving those dynamics? Does the geographic mosaic of coevolution shape adaptation in fundamentally different ways in different forms of interaction? How does the geographic mosaic of coevolution shape speciation? How does the structure of reciprocal selection change during the assembly of large webs of interacting species? How important are genomic events such as whole-genome duplication (i.e., polyploidy) and whole-genome capture (i.e., hybridization) in generating novel webs of interacting species? I end by suggesting four points about coevolution that we should tell every new student or researcher in biology.