In this talk, we overview recent developments for quantitative analysis of asymptotic behavior of energy solutions to the Cauchy--Dirichlet problem for fast diffusion equations posed in bounded domains. It is well known that every energy solution vanishes in finite time and a suitably rescaled solution converges to an asymptotic profile, which is a nontrivial solution for a semilinear elliptic equation. Bonforte and Figalli (CPAM, 2021) first determined an exponential rate of convergence to nondegenerate positive asymptotic profiles for nonnegative rescaled solutions in a weighted $L^2$ norm for smooth (at least $C^2$) bounded domains by developing the so-called nonlinear entropy method. On the other hand, the speaker (ARMA, 2023) developed an energy method along with a quantitative gradient inequality and also proved the same exponential convergence in the Sobolev norm for bounded $C^{1,1}$ domains. The optimality of the exponential rate was conjectured in view of some formal linearized analysis; however, it was not proved due to some difficulty arising from nontrivial stability nature of asymptotic profiles in the fast diffusion setting. Furthermore, the nondegeneracy of asymptotic profiles was indispensable in these works. In this talk, these results are extended to possibly sign-changing (nondegenerate) asymptotic profiles as well as general bounded domains by improving the energy method as well as quantitative gradient inequality. Moreover, the optimality of the exponential rate is also proved. Furthermore, such a quantitative analysis is also extended to degenerate asymptotic profiles. This talk is based on recent joint works with Yasunori Maekawa (Kyoto University) and Norihisa Ikoma (Keio University).