This paper introduces to readers the new concept and methodology of confidence distribution and the modern-day distributional inference in statistics. This discussion should be of interest to people who would like to go into the depth of the statistical inference methodology and to utilize distribution estimators in practice. We also include in the discussion the topic of generalized fiducial inference, a special type of modern distributional inference, and relate it to the concept of confidence distribution. Several real data examples are also provided for practitioners. We hope that the selected content covers the greater part of the developments on this subject.

To appear as a chapter in Springer Handbook of Engineering Statistics, 2nd ed