We did this at CloudSploit. Our entire service is written using Lambda-based APIs behind the API Gateway. The biggest challenge at first was the cold boot times. If our API went un-pinged for more than five minutes the next request would take > 5 seconds (!). This was solved with a CloudWatch event that invokes the function every 4 seconds (which is silly in my mind, but it works).

You can read more about our process here: https://blog.cloudsploit.com/we-made-the-whole-company-serve...