Significant developments in non-parametric and semi-parametric regression methods for longitudinal data have taken place in the last 10 years. The presence of the within-subject correlation among repeated measures over time presents major challenges in developing

kernel and spline smoothing methods for longitudinal data. Classical local likelihood based kernel methods and their natural local estimating equation extensions fail to effectively account for the within-subject correlation. This difficulty has called for the development of a new class of non-local kernel estimators. Extension of spline smoothing to longitudinal data requires explicitly accounting for the within-subject correlation in constructing the penalized likelihood function. Statistical theory has recently been developed to understand the properties of these advanced kernel and spline methods for longitudinal data. In this chapter we provide a review of both estimating equation based methods and likelihood based methods for non-parametric and semi-parametric regression using kernel and spline smoothing for longitudinal data. Connections between splines (smoothing splines and Psplines) and mixed models are discussed. More detailed discussions about this connection can also be found in Chapter 11 and Chapter 12.