The vulnerability against interference, spoofing, and jamming of GNSS receivers is considered nowadays a major security concern. This security threat is exacerbated with the existing market availability of GPS jamming and spoofing equipment sold at reasonable prices. If jamming is the main issue faced at present, spoofing, which allows hijacking someone from the expected path, may lead to even worse consequences. Even with the latest security measures that are going to be deployed on the Galileo PRS signals, GNSS receivers are prone to attacks that are relatively easy to implement. In this paper, we identify different countermeasures and security schemes that can be used against spoofing attacks. These countermeasures include some modifications on the GNSS receiver's side, rather than requiring modifications of the whole existing GNSS infrastructure. More specifically, we propose a detection and protection scheme consisting of several statistical tests, based on the computations of moving variances of Doppler offset and C/No estimates, together with a consistency test of the PVT computation. We evaluate the performance of the proposed scheme through simulations and using a measurement setup consisting of a Spirent GSS8000 full constellation simulator whose output is combined with the one from a rooftop GPS antenna before being fed to a receiver front-end. Finally, we compute the probability of detection and false alarm in spoofing detection using the proposed scheme.