This book examines in detail the correlation, specifically the weighted correlation, and applications involving rankings. A general application is the evaluation of methods to predict rankings. Other applications involve rankings representing human preferences to infer user preferences, the use of weighted correlation with microarray data, and those in the domain of time series. In this book, we present new weighted correlation coefficients and new methods of weighted principal component analysis. We also introduce new methods of dimension reduction and clustering for time series data and describe some theoretical results on the weighted correlation coefficients in separate sections.