What guarantees are there on the run-time complexity (Big-O) of LINQ methods?

There are very, very few guarantees, but there are a few optimizations:

• Extension methods that use indexed access, such as ElementAt, Skip, Last or LastOrDefault, will check to see whether or not the underlying type implements IList<T>, so that you get O(1) access instead of O(N).

• The Count method checks for an ICollection implementation, so that this operation is O(1) instead of O(N).

• Distinct, GroupBy Join, and I believe also the set-aggregation methods (Union, Intersect and Except) use hashing, so they should be close to O(N) instead of O(N²).

• Contains checks for an ICollection implementation, so it may be O(1) if the underlying collection is also O(1), such as a HashSet<T>, but this is depends on the actual data structure and is not guaranteed. Hash sets override the Contains method, that’s why they are O(1).

• OrderBy methods use a stable quicksort, so they’re O(N log N) average case.

I think that covers most if not all of the built-in extension methods. There really are very few performance guarantees; Linq itself will try to take advantage of efficient data structures but it isn’t a free pass to write potentially inefficient code.