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Diffusion tensor imaging (DTI) provides important information on the structure of white matter fiber bundles as well as detailed tissue properties along these fiber bundles in vivo. This paper presents a functional regression framework, called FRATS, for the analysis of multiple diffusion properties along fiber bundle as functions in an infinite dimensional space and their association with a set of covariates of interest, such as age, diagnostic status and gender, in real applications. The functional regression framework consists of four integrated components: the local polynomial kernel method for smoothing multiple diffusion properties along individual fiber bundles, a functional linear model for characterizing the association between fiber bundle diffusion properties and a set of covariates, a global test statistic for testing hypotheses of interest, and a resampling method for approximating the p-value of the global test statistic. The proposed methodology is applied to characterizing the development of five diffusion properties including fractional anisotropy, mean diffusivity, and the three eigenvalues of diffusion tensor along the splenium of the corpus callosum tract and the right internal capsule tract in a clinical study of neurodevelopment. Significant age and gestational age effects on the five diffusion properties were found in both tracts. The resulting analysis pipeline can be used for understanding normal brain development, the neural bases of neuropsychiatric disorders, and the joint effects of environmental and genetic factors on white matter fiber bundles.