These notes are based on a regional set of lectures on curve estimation in the context of independent and dependent observations given at the University of California, Davis during June 1989. Much of these lectures is concerned with probability density or regression function estimation when observations are independent. The character of the asymptotic results (at least locally) is qualitatively the same if the observations are those of a dependent stationary sequence with short-range dependence. Research in the case of long-range dependence is recent and qualitatively of a different character. One should note that the spectral estimation (also discussed here) is of an older vintage and results there in part motivated later research on probability density estimates. Many of the results have the imprint both of stochastic methods as well as those of approximation theory.