This book focuses on correlation coefficients and its applications in applied science fields. The book begins by describing the historical development and various types of correlations.  Rank correlation methods including Pearson's, Spearman's, and Kendall's correlation are discussed at length. The book also discusses sampling distribution of correlation coefficients and applications of correlations in various fields. The book presents novel topics such as (i) a quick analytical method to approximate Pearson's correlation, (ii) single-variable correlation, (iii) fractional co-skewness and co-kurtosis, and (iv) the fallacy on correlation between the sample mean and sample variance.  This book is ideal for courses on mathematical statistics, engineering statistics, and exploratory data analysis and is primarily aimed at upper-undergraduate and graduate level students.  The book is also useful for researchers and professionals in various fields who are interested in data analysis.