One of the most widespread approaches to reactive planning is Schoppers' universal plans. We propose a stricter definition of universal plans which guarantees a weak notion of soundness, not present in the original definition, and isolate three different types of completeness that capture different behaviors exhibited by universal plans. We show that universal plans which run in polynomial time and are of polynomial size cannot satisfy even the weakest type of completeness unless the polynomial hierarchy collapses. By relaxing either the polynomial time or the polynomial space requirement, the construction of universal plans satisfying the strongest type of completeness becomes trivial. As an alternative approach, we study randomized universal planning. By considering a randomized version of completeness and a restricted (but nontrivial) class of problems, we show that there exists randomized universal plans running in polynomial time and using polynomial space which are sound and complete for the restricted class of problems. We also report experimental results on this approach to planning, showing that the performance of a randomized planner is not easily compared to that of a deterministic planner.