In this short review, we will briefly discuss the story of hydrogen storage, its impact on clean energy application, especially the challenges of using hydrogen adsorption for onboard application. After a short comparison of the main methods of hydrogen storage (high pressure tank, metal hydride and adsorption), we will focus our discussion on adsorption of hydrogen in graphitic carbon based large surface area adsorbents including carbon nanotubes, graphene and metal organic frameworks. The mechanisms, advantages, disadvantages and recent progresses will be discussed and reviewed for physisorption, metal-assisted storage and chemisorption. In the last section, we will discuss hydrogen spillover chemisorption in detail for the mechanism, status, challenges and perspectives. We hope to present a clear picture of the present technologies, challenges and the perspectives of hydrogen storage for the future studies.