In mean-unbiased estimation we normally prefer an estimator, in the class of mean-unbiased estimators, which minimizes the variance. The classical Cramer-Rao inequality provides a sometimes-achievable lower bound for the variance of a mean-unbiased estimator, which can be useful in assessing a mean-unbiased estimator.;Building on the work of Alamo and Stangenhaus and David, in this dissertation we develop analogues of the Cramer-Rao inequality for median-unbiased estimators having absolutely continuous distributions, based on certain measures of dispersion which we call diffusivity. Three kinds of the diffusivity are defined. These measures are natural measure of dispersion associated with median-unbiased estimators, but diffusivity is different from conventional measures of dispersion or spread in that it measures vertical spread of a density rather than horizontal spread.;When we have distribution depending upon a single real-valued parameter, the analogues of the Cramer-Rao inequality for median-unbiased estimators show the same pattern as the Cramer-Rao inequality itself, with the Fisher information and the variance in the Cramer-Rao inequality replaced by the first absolute moment of the sample score and the diffusivity of the median-unbiased estimator, respectively. An analogue of the Chapman-Robbins inequality which is free from regularity conditions is also given.;We identify optimal median-unbiased estimators of the location and scale parameters for special location and scale families of distributions.;The analogues of the Cramer-Rao inequality are generalized to the vector-valued parameter case, in which the resulting lower bound can be regarded as an analogue of the bound for the generalized variance for mean-unbiased estimation.;The role of mode-unbiasedness in median-unbiased estimation is discussed. An extension of the inequalities to multivariate distributions is given.