I'm not sure if they're turing complete, however many many many common regexp dialects (as seen in perl, ruby, python, etc.) are more powerful than what you learned as "regular expressions" in CS class, they're at least as powerful as a PDA.
Most regex implementations allow you to parse some context-free grammars, and even some non-context-free grammars. But they don't let you parse all context-free grammars.
Him: Do you know how to check if it's possible to write a regular expression for this?
Me: Either create an automata and then it's demonstrably possible, or apply the pumping lemma to prove it's impossible?
Him: No. Dare Stackoverflow to write it.