This article reviews some second-order approximations to the posterior mean, and the marginal distribution of observations called a marginal likelihood, where “second-order” means that the approximation has asymptotic errors of order
O
(
n
−
2
)
O(n^{-2})
as the sample size
n
n
goes to infinity. In addition, the Laplace approximations using asymptotic modes instead of the exact mode of integrand, and higher-order asymptotic expansions are presented.