Normal distribution is found in many applications and plays a particularly important role in applied problems. The theoretical justification for the exclusive role of the normal distribution is provided by the limit theorems of probability theory. The qualitatively corresponding result of this can be explained as follows. The normal distribution is a good approximation whenever the random variable under study is the sum of many independent random variables, the largest of which is small compared to the total sum. Univariate Normal Distribution with Applications is devoted in general to the description of the different properties of normally distributed random variables. In Chapter 1, the list of the main probability distributions is given. Chapter 2 is devoted to the definitions and the basic properties of the family of normal random values. The different characteristics of these random variables are discussed in Chapter 3—sometimes we need to work with the ordered random observations. The ordered normal observations, their distributions and properties are considered in Chapter 4. Descriptions of the different options of the CLT are given in Chapter 5. In Chapter 6, characterizations of normal distribution under various conditions are presented. Lastly, in Appendix, the main properties and characteristics of the different random variables and their properties mentioned in the text are discussed.