Many social networks and complex systems are found to be naturally divided into clusters of densely connected nodes, known as community structure (CS). Finding CS is one of fundamental yet challenging topics in network science. One of the most popular classes of methods for this problem is to maximize Newman's modularity. However, there is a little understood on how well we can approximate the maximum modularity as well as the implications of finding community structure with provable guarantees. In this paper, we settle definitely the approximability of modularity clustering, proving that approximating the problem within any (multiplicative) positive factor is intractable, unless P = NP. Yet we propose the first additive approximation algorithm for modularity clustering with a constant factor. Moreover, we provide a rigorous proof that a CS with modularity arbitrary close to maximum modularity QOPT might bear no similarity to the optimal CS of maximum modularity. Thus even when CS with near-optimal modularity are found, other verification methods are needed to confirm the significance of the structure.