How? Not very much. The issue is that cloud is expensive, and often does't fit the budget rules for grants. After all, cloud providers are in business to extract as much profit from clients as possible. Amazon's rates, for instance, basically sell you the computer each year. Academics often have much lower-overhead resources available (campus HPC centers, idle grad students, etc).
Almost everyone uses cloud in a SaaS sense, though: dropbox, gmail, etc. There's nothing different about academics.
Google apps for education is quite useful for academic administration through the use of forms. The software's functionality can be extended by adding scripts (javascript) or plugins.
The same forms can be used by research projects to collect questionaire data.
Mendeley is a good tool for reference management with cloud backup and sharing options. The institutional edition allows groups of students or researchers to efficiently work together to build a literature database on their research topic.
As for the infrastructure as a service, Amazon may indeed seem expensive for big institutions, but offers good service for smaller colleges that outsource most of their IT infrastructure. I use it to teach advanced IT courses to small groups of students (less then 10 students). Students often have their own laptops with various hardware and software configurations. Some may not be able to run multiple virtual machines efficiently on their local machine. Giving them access to standardised, pre-configured machines on Amazon was found to be an efficient solution and students can get an amount of free time when they sign up for the first time to Amazon Web Services.
But Amazon is not the only provider: Swiss users may want to look at https://www.cloudsigma.com if they need to keep sensitive data stored inside the country. (I have no relationship with them)
From my field interaction with some customers, academicians in universities generally deploy cloud computing in smaller scale for courses teaching, usage pilot test & research experimentation especially based on OpenSource like OpenStack, CloudStack, OpenNebula etc. Rationale is these platforms are cheaper than commercial platform like VMware vSphere & Microsoft Hyper-V etc. Some also testing out public cloud SaaS for some non-critical apps & to evaluate security vulnerability.
Some academics / universities also called-in by government agencies in-charged on IT policy & roll-out to deploy IaaS & PaaS clouds to other government agencies to evaluate how effective & secured the cloud computing is before big bang / phase deployment.