Applied interest in considering nonlinear structural equation models has increased in recent years. This dissertation deals with fitting a nonlinear structural equation model consisting of two parts: a linear measurement model relating observed variables to unobserved concepts or factors, and a nonlinear structural model where one particular factor is a polynomial in other factors. Two new statistical procedures are developed, and their justification and usefulness are discussed;The first procedure follows the ad-hoc approach that has been used in some applications. Some statistical limitation is revealed for the existing approach, and a new general method called the generalized appended product indicator (GAPI) procedure is introduced. The GAPI procedure uses products of observed variables as indicators for nonlinear terms in the structural model. Then, a proper model is fitted to the combined set of indicators by minimizing a discrepancy function. The resulting estimator is consistent without assuming any distributional form for the underlying factors or errors. Issues regarding identifiability and standard error estimation are discussed. A simulation study addresses statistical issues including comparisons of discrepancy functions and the choice of appended product indicators. One advantage of the GAPI procedure is that it can be implemented using existing software;An alternative approach developed here is the two-stage method of moment (2SMM) procedure which is applicable for virtually any polynomial structural modeling problem. In the first stage, the measurement model is fitted, and factor score estimates are obtained along with estimated moments of the measurement error associated with the factor score estimation. In the second stage, the nonlinear structural model parameter is estimated based on the factor score estimates by applying a method of moments procedure similar to that used in the errors-in-variables regression. The asymptotic properties of the 2SMM estimator are derived, and a modified estimator with better small sample properties is introduced. The asymptotic covariance matrix of the estimator which incorporates variability from the first stage estimation is presented. Various simulation studies are presented showing the superiority of 2SMM over other methods. An example from a substance-abuse prevention study is also discussed.