Opportunistic use of unused spectrum could efficiently be carried out using the paradigm of Cognitive Radio (CR). A spectrum remains idle when the primary user (licensee) is not using it. The secondary nodes detect this spectral hole quickly and make use of it for data transmission during this interval and stop transmitting once the primary starts transmitting. Detection of spectral holes by the secondary is called spectrum sensing in the CR scenario. Spectrum Sensing is formulated as a hypothesis testing problem wherein under H0 the spectrum is free and under H1, occupied. The samples will have different probability distributions, P0 and P1, under H0 and H1 respectively. In the first part of the thesis, a new algorithm - entropy test is presented, which performs better than the available algorithms when P0 is known but not P1. This is extended to a distributed setting as well, in which different secondary nodes collect samples independently and send their decisions to a Fusion Centre (FC) over a noisy MAC which then makes the final decision. The asymptotic optimality of the algorithm is also shown. In the second part, the spectrum sensing problem under impediments such as fading, electromagnetic interference and outliers is tackled. Here the detector does not possess full knowledge of either P0 or P1. This is a more general and practically relevant setting. It is found that a recently developed algorithm (which we call random walk test) under suitable modifications works well. The performance of the algorithm theoretically and via simulations is shown. The same algorithm is extended to the distributed setting as above.