Estimators of the form $\hat f_n(x) = (1/n) \sum^n_{i=1} \delta_n(x - x_i)$ of a probability density $f(x)$ are considered, where $x_1 \cdots x_n$ is a sample of $n$ observations from $f(x)$. In Part I, the properties of such estimators are discussed on the basis of their mean integrated square errors $E\lbrack\int(f_n(x) - f(x))^2dx\rbrack$ (M.I.S.E.). The corresponding development for discrete distributions is sketched and examples are given in both continuous and discrete cases. In Part II the properties of the estimator $\hat f_n(x)$ will be discussed with reference to various pointwise consistency criteria. Many of the definitions and results in both Parts I and II are analogous to those of Parzen [1] for the spectral density. Part II will appear elsewhere.