How established are ORMs (object relational mapping) in the world of databases

A lot of places are using them, that doesn't mean they are using them well or that they are a good idea for the long term health of the database. Doesn't mean they aren't either, usually it just means the people choosing them don't think about how this affects database design, maintenance and performance over time.

ORMs are widely used. Django (python web application framework) uses SQLAlchemy. Hibernate is popular for Java programs. ORMs make SQL application development faster, reduce the amount of boilerplate code that you must write, and hide which database you are using from the rest of the application. Performance may suffer using an ORM, so be prepared to customize things as needed.

Widely used and definitely the present and near future. Database access through a handcoded layer of SQL generation was always fraught with drudgery and typos, and was unwieldy at best. ORMs let you use a persistence store in a programming way.

I thought this blog argued for it well: http://jonkruger.com/blog/category/fluent-nhibernate/ and SO posts like this (nHibernate versus LLBLGen Pro) show just how many people are using them.