Google App Engine is Google Cloud's managed, serverless application platform for building and running web applications and APIs. Developers supply code, and Google handles infrastructure, load balancing and scaling. There are two environments: the standard environment, in which instances run in a sandbox and can scale down to zero, and the flexible environment, in which Docker containers run on Compute Engine virtual machines. Google's own documentation advises new users to consider Cloud Run as the preferred alternative.
App Engine is provided by Google LLC, headquartered in Mountain View, California and wholly owned by Alphabet, and is subject to US jurisdiction, including the CLOUD Act. When you create an app you choose a region, and that choice cannot be changed afterwards. A European region is available, with a default bucket in the EU multi-region. Even with storage in the EU, the provider's US jurisdiction remains relevant for a European organisation. Google publishes a GDPR page for Google Cloud; check the terms in your own contract.