This thesis discusses the current screening algorithm that is used to detect fetal Down's syndrome. The algorithm combines a model for predicting age related risks and a model for appropriately transformed serum concentrations to produce estimates of risks. A discriminant analysis is used to classify pregnancies as either unaffected or Down's syndrome. The serum concentrations vary with gestational age and the relationship between serum concentrations and gestational age is modelled using regression. These models are discussed and alternative models for these relationships are offered. Concentration values are generally expressed in terms of multiples of the medians for unaffected pregnancies, or MoM values, which involves grouping the concentrations into weekly bins. Transformations of the MoM values are used in the model for predicting risks. The transformed values are equivalent to the residuals of the fitted regression models. This thesis directly models the residuals rather than converting the data to MoM values. This approach avoids the need to group gestational dates into completed weeks. The performance of the algorithm is assessed in terms the detection rates and false positive rates. The performance rates are prone to considerable sampling error. Simulation methods are used to calculate standard errors for reported detection rates. The bias in the rates is also investigated using bootstrapping techniques. The algorithm often fails to recognize abnormalities other than Down's syndrome and frequently associates them with low risks. A solution to the problem is offered that assigns an index of atypicality to each pregnancy, to identify those pregnancies that are atypical of unaffected pregnancies, but are also unlike Down's syndrome pregnancies. Nonparametric techniques for estimating the class conditional densities of transformed serum values are used as an alternative to the conventional parametric techniques of estimation. High quality density estimates are illustrated and these are used to compute nonparametric likelihood ratios that can be used in the probability model to predict risks. The effect of errors in the methods of recording gestational dates on the parameter estimates that are used in the discriminant analysis is also considered.