In the estimation of growth models the two main problems are a) they are oftennon-linear and b) the errors are serially correlated. Methodology for non-linear growth models with serially correlated errors usig the mixed models approach are available in the literature (Lindstrom and Bates (1990)). These methods are mainly useful for estimating population averages of growth from data that are observed over short time periods. In thisdissertation a method for estimating individual specific non-linear growth curves under first order auto-regressive (AR(1)) error structure for data that are observed over a long period of time is presented. In order to reduce the computational complexity the maximum likelihood estimation procedure is separated into two parts using a quasi-likelihood method called here themarginal likelihood method. In this method the AR(1)‘nuisance’parameter is estimated from a marginal likelihood that contains the AR(1) parameter alone, under a polynomial approximation of the non-linear model. Then the non-linear model parameters are estimated by assuming the AR(1) parameter to be known, where the known value is the estimate. Simulation results are presented to discuss the properties of the properties of the marginal likelihood estimation method and to test the validity of the polynomial approximation. The method is also compared with the Restricted Maximum Likelihood Estimation (REML) proposed under a mixed model structed by Harville(1977). The method is applied to two data sets from maternal and child health studies. These data on fundal heights are analyzed using the marginal likelihood approach and compared with other approaches.