Many studies yield functional data, with the ideal units of observation curves and observed data sampled on a fine grid. These curves frequently have irregular features requiring spatially adaptive nonparametric representations. I will discuss new methods for modeling these data using functional mixed models, which treat the curves as responses and relate them to covariates using nonparametric fixed and random effect functions. The Bayesian wavelet-based approach yields adaptively regularized posterior samples for all model parameters that can be used for any desired Bayesian estimation, inference, or prediction. I will illustrate this method on several applications yielding spiky functional data, with special focus on MALDI-TOF mass spectrometry proteomics data. Time allowing, I will also describe how it can be extended to deal with incomplete functional data for which some regions of some functions are missing and to model higher-dimensional functional data (e.g., images).