This article is only a very general and theoretical introduction to the modified normal distributions. The author is presently engaged in further development of the theory in the analysis of various examples, the most remarkable of which are of astronomical origin. The lineo-normal distributions seem to be strikingly well confirmed by many types of observations and measurements. It is only through the analysis of a great number of large samples that the concept of zero error and its impact on the general theory of errors will be definitely clarified.