Acquisition and tracking systems form an important component of free-space optical communications due to directional nature of the optical signal. Acquisition subsystems are needed in order to search and locate the receiver terminal in an uncertainty/search region with very narrow laser beams. In this paper, we have proposed and analyzed two adaptive search schemes for acquisition systems that perform better---for the low probability of detection---than the spiral scanning approach. The first of these schemes, the adaptive spiral search, provides a better acquisition time performance by dividing the search region into a number of smaller subregions, and prioritizing search in regions of higher probability mass. The second technique---the shotgun approach---searches the region in a random manner by sampling the search region according to a Gaussian distribution. The adaptive spiral scheme outperforms the shotgun approach in terms of acquisition time, especially if the number of search subregions is large enough. However, a higher pointing accuracy is required by the adaptive spiral search in order to search the region precisely. On the other hand, the shotgun scanning approach does not require such stringent pointing accuracy.