A Metropolis-Hastings Robbins-Monro (MH-RM) algorithm is proposed for maximum likelihood estimation in a general nonlinear latent structure model. The MH-RM algorithm represents a synthesis of the Markov chain Monte Carlo method, widely adopted in Bayesian statistics, and the Robbins-Monro stochastic approximation algorithm, well known in the optimization literature. The general latent structure model not only encompasses linear structural equations among latent variables, but also includes provisions for nonlinear latent regressions. Based on item response theory, a comprehensive measurement model provides the link between the latent structure and the observed variables. The MH-RM algorithm is shown to converge to a local maximum of the likelihood surface with probability one. Its significant advantages in terms of flexibility and efficiency over existing algorithms are illustrated with applications to real and simulated data. Implementation issues are discussed in detail. In addition, this dissertation integrates research on the parametrization and estimation of complex nonlinear latent variable models and furthers the understanding of the relationship between latent trait models and incomplete data estimation.