I was recently asked to evaluate Lambda for use in hosting a bunch of existing web sites, and just from README.md, I'm glad I backed the hell away from that pandora's box before opening it.
Surprised WWW-Authenticate headers aren't passed through. May be possible with custom authorizers[3], though it's extra complexity.
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[1] https://aws.amazon.com/blogs/compute/binary-support-for-api-...
[2] https://github.com/serverless/serverless/issues/2797
[3] http://docs.aws.amazon.com/apigateway/latest/developerguide/...
Lambdas are great for anything you can wire them up to in AWS, but using them with API Gateway is a pain.
For things like processing files or images as they appear in an S3 bucket, feeding or eating from a queue, replacing CRON jobs or small ETLs, and about a dozen other uses I have found them to be amazing.