The paper investigates the computational problem of predicting RNA secondary structures. The general belief is that allowing pseudoknots makes the problem hard. Existing polynomial-time algorithms are heuristic algorithms with no performance guarantee and can only handle limited types of pseudoknots. In this paper we initiate the study of predicting RNA secondary structures with a maximum number of stacking pairs while allowing arbitrary pseudoknots. We obtain two approximation algorithms with worst-case approximation ratios of 1/2 and 1/3 for planar and general secondary structures,respectively. For an RNA sequence of $n$ bases, the approximation algorithm for planar secondary structures runs in $O(n^3)$ time while that for the general case runs in linear time. Furthermore, we prove that allowing pseudoknots makes it NP-hard to maximize the number of stacking pairs in a planar secondary structure. This result is in contrast with the recent NP-hard results on psuedoknots which are based on optimizing some general and complicated energy functions.