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The Regex class itself is thread safe and immutable (read-only). That is, Regex objects can be created on any thread and shared between threads; matching methods can be called from any thread and never alter any global state.
However, result objects (Match and MatchCollection) returned by Regex should be used on a single thread. Although many of these objects are logically immutable, their implementations could delay computation of some results to improve performance, and as a result, callers must serialize access to them.
If there is a need to share Regex result objects on multiple threads, these objects can be converted to thread-safe instances by calling their synchronized methods. With the exception of enumerators, all regular expression classes are thread safe or can be converted into thread-safe objects by a synchronized method.
Enumerators are the only exception. An application must serialize calls to collection enumerators. The rule is that if a collection can be enumerated on more than one thread simultaneously, you should synchronize enumerator methods on the root object of the collection traversed by the enumerator.