This dissertation, composed of three papers to be submitted for publication in scholarly journals, focuses on Bayesian methods in function estimation. Chapter 2 specifically discusses spectral density estimation. The semiparametric estimator derived in this chapter combines a smoothed version of the periodogram with a parametric estimator of the spectral density. This semiparametric estimator, which shrinks towards the parametric form provided it is correct, is derived from a hierarchical model. This estimator is consistent, it is competitive with other estimators (as seen through simulation studies), and ultimately does not require the specification of a parametric form.;The third and fourth chapters begin by modeling longitudinal data with linear mixed regression splines. The knots associated with the fixed and random effect curves (in the mixed model) are identified using Bayesian methods. In Chapter 3, reversible jump MCMC methods are used to sample from the marginal posterior of the knots associated with these two curves. Sampling from such a posterior, however, requires evaluation of the marginal likelihood of the knots. This marginal likelihood can not be calculated. Two sampling methods are thus considered in this chapter; these two methods correspond to two different approximations of this likelihood and are compared on how effectively they penalize models with unnecessarily large values of random effect knots.;In the fourth chapter, a similar posterior is considered. This posterior, however, relies on the decomposition of the random effect curve into orthogonal principal component curves, and restricts the random effect curves to have the same knots as the fixed effect curve. The knots associated with the fixed and random effect curves and the number of significant principal component curves is identified by sampling from their joint posterior distribution of knots.