The study of the spectral gap is motivated from three aspects: The phase transitions, the reandom algorithm and the spectrum theory. Then we survey the main mathematical tools used in the study in the past years, including some geometric, analytic and probabilistic methods. Especially, we mention a couple of new variational formulas for the lower bound of the spectral gap. Finally, we report a generalized Cheeger's inequality (originally coming from Riemannian geometry) for general symmetric forms and an existence criterion for the spectral gap of general reversible Markov processes.