Can you run Hadoop in the cloud?

As a solutions architect at Pythian, I often get questions from clients about the many solutions available to them to address their big data needs. Between Hadoop, cloud-based, and hybrid solutions, finding the best option for their unique needs can be a daunting challenge.
Sometimes Hadoop is the answer, and sometimes a cloud solution is a better fit. Determining the right direction requires us to consider multiple factors including the specific need, use case(s), budget, available resources within the organization and data volume, to name a few.
With this post, I’m going to try to answer some of the most common questions presented to us by clients, including: At what point is big data big enough to require a system like Hadoop? At what point is it wise to run big data workloads on the cloud? What are its limitations? How does it compare to cloud? And can you run Hadoop on the cloud?