It has been long proposed that the brain should perform computation efficiently to increase the fitness of the organism. However, the validity of this prominent hypothesis remains largely debated. I have investigated how the idea of efficient computation can guide us to understand the operational regimes underlying various functions of the brain, in particular in the domain of perception and spatial cognition. In the first line of research, I demonstrate that such idea leads to a well-constrained yet powerful model framework for human perceptual behaviors by assuming the system is efficient both in term of encoding and decoding. This framework, when applying to human visual perception, explains many reported perceptual biases, including the repulsive biases away from the prior expectation, which are counter-intuitive according to the traditional Bayesian view. This framework also predicts that two basic psychophysical measures, i.e., perceptual bias and discrimination threshold, should be directly linked via a simple equation. This predicted relation is well supported by a large array of published data. In the second line of research, I demonstrate that a theory based on efficient coding makes quantitative predictions on the functional architecture of the grid cell system in rodents. One such prediction is that the spatial scales of grid modules should follow a geometric progression,and furthermore the scaling factor should lie robustly between the range of 1.4 to 1.7. These predictions closely match the data reported in recent neurophysiological experiments. Together, these results suggest that achieving efficient computation may serve as a basic computational principle which generalizes across neural systems processing low-level and high-level functions.