The recent growing interest for indoor Location-Based Services (LBSs) has created a need for more accurate and real-time indoor positioning solutions. The sparse nature of location finding makes the theory of Compressive Sensing (CS) desirable for accurate indoor positioning using Received Signal Strength (RSS) from Wireless Local Area Network (WLAN) Access Points (APs). We propose an accurate RSS-based indoor positioning system using the theory of compressive sensing, which is a method to recover sparse signals from a small number of noisy measurements by solving an \ell_1-minimization problem. Our location estimator consists of a coarse localizer, where the RSS is compared to a number of clusters to detect in which cluster the node is located, followed by a fine localization step, using the theory of compressive sensing, to further refine the location estimation. We have investigated different coarse localization schemes and AP selection approaches to increase the accuracy. We also show that the CS theory can be used to reconstruct the RSS radio map from measurements at only a small number of fingerprints, reducing the number of measurements significantly. We have implemented the proposed system on a WiFi-integrated mobile device and have evaluated the performance. Experimental results indicate that the proposed system leads to substantial improvement on localization accuracy and complexity over the widely used traditional fingerprinting methods.