Longitudinal functional data consist of functional data collected at multiple time points for which the observational times may vary by subject. They differ from traditional longitudinal data in that the observation at each time point is a function rather than a scalar. The aim is to extend the traditional linear mixed-effects model for longitudinal data to longitudinal functional data. We study a class of functional mixed effects models(FMEM), which includes fixed effects that characterize the association between longitudinal functional responses and covariates of interest and random effects that capture the intricate correlation structure of longitudinal functional responses. We propose local linear estimates for the fixed-effect coefficient functions and establish their asymptotic properties. We also develop a simultaneous confidence band for each fixed-effect coefficient function and a global test for linear hypotheses of these coefficient functions. The numerical performance of the proposed methods is examined through an extensive simulation study and an application to white-matter fiber data from a national database for autism research.