I applied online. The process took 2 weeks. I interviewed at Amazon (Seattle, WA) in Nov 2013
Interview
Two phone interviews before on-site interview.
The first asked about data structure and resume based questions.
The second asked about machine learning method, including linear regression assumptions, cluster analysis, bayesian method.
The whole process is amazingly fast. They'll let you know the second after each interview.
Interview questions [1]
Question 1
Asked about Ito's formula as stochastic calculus is on the resume, although it's totally unrelated to the job.