To discover and summarize regular periodic variation in a time series, $\{ y_i ,t_i \} $, we may consider approximating the observed $y_i $ with a general smooth periodic function of $t_i $:\[ y_i \approx f \left( {\omega t_i } \right) \]where $f( \cdot )$ is a periodic function with period 1 (so that $f(\omega t)$ has period $\lambda = {1 / \omega }$ as a function of t). To make such an approximation, we need to choose a function $f( \cdot )$ the frequency, $\omega $, may be known, or it may need to be chosen also. The basic suggestion presented in this paper is to choose the function, $f( \cdot )$, by smoothing $y_i $ as a function of $(\omega t_i \bmod 1)$. If $\omega $ is unknown, it can be chosen to make $f(\omega t_i )$ a good approximation to $y_i $. More complex periodic variation can be modelled using a variation of the Projection Pursuit Paradigm (Friedman and Stuetzle (1982a)) to construct approximations of the form\[ y_i \approx \sum_{k = 1}^K {f_k \left( {\omega _k t_i } \right)} . \] This approach has been developed as part of the Orion project for statistical graphics at the Stanford Linear Accelerator Center; it is most useful in an interactive graphics environment, like the Orion I workstation (Friedman and Stuetzle (1981b), McDonald (1982a, b, c)). However, given moderate compromises, it can be used with profit in more conventional environments for statistical computing. Some advantages of periodic smoothing are: The observed times, ti, need not be equally spaced, which means, in particular, that missing data is not a problem. The function, $f( \cdot )$, and the frequency, $\omega $, can be chosen in a way that is insensitive to occasional gross errors in $y_i $ (and ti, for that matter). The function, $f( \cdot )$, need not be easily approximated by simple linear combinations of sines and cosines at harmonics of the fundamental frequency, $\omega $. Perhaps most importantly, the model can be constructed and modified interactively, and has a natural graphical representation.