Hydrogen spillover refers to catalytic dissociation of hydrogen molecules followed by surface diffusion to the catalytic support. This mechanism has been proposed as a means to initiate hydrogen storage at ambient temperature. When catalytic dissociation of hydrogen is used to initiate room temperature hydrogen storage, this initiates weak chemisorption. High variations in experimental uptake, combined with theoretical calculations that are inconsistent with this mechanism, have made this concept controversial. Here, we review hydrogen uptake in porous carbon and metal-organic frameworks (MOFs), with emphasis on spectroscopic evidence of atomic hydrogen bound to the surface and resolution of discrepancies between experimental and theoretical studies. We conclude with a perspective of hydrogen spillover for future material design, hydrogen storage, and fundamental understanding of a process that is crucial for adsorption and catalysis.