The problem of unbiased estimation, restricted only by the postulate of section 2, is considered here. For a chosen number $s > 1$, an unbiased estimate of a function $g$ on the parameter space, is said to be best at the parameter point $\theta_0$ if its $s$th absolute central moment at $\theta_0$ is finite and not greater than that for any other unbiased estimate. A necessary and sufficient condition is obtained for the existence of an unbiased estimate of $g$. When one exists, the best one is unique. A necessary and sufficient condition is given for the existence of only one unbiased estimate with finite $s$th absolute central moment. The $s$th absolute central moment at $\theta_0$ of the best unbiased estimate (if it exists) is given explicitly in terms of only the function $g$ and the probability densities. It is, to be more precise, specified as the l.u.b. of certain set $\mathcal{a}$ of numbers. The best estimate is then constructed (as a limit of a sequence of functions) with the use of only the data (relating to $g$ and the densities) associated with any particular sequence in $\mathcal{a}$ which converges to the l.u.b. of $\mathcal{a}$. The case $s = \infty$ is considered apart. The case $s = 2$ is studied in greater detail. Previous results of several authors are discussed in the light of the present theory. Generalizations of some of these results are deduced. Some examples are given to illustrate the applications of the theory.