This textbook provides a well-motivated introduction to central and diverse topics, methods, and applications of mathematical statistics. Starting from statistical data analysis, it addresses and justifies classical as well as newer construction principles for statistical procedures. The book aims to introduce not only classical areas but also newer applications. These range from methods of asymptotic statistics to nonparametric estimation methods, robust and sequential tests, as well as the statistics of counting processes with significant applications, such as in survival analysis, to image processing and reconstruction, and to quantile hedging in financial mathematics. The book demonstrates that mathematical statistics is a field rich with particularly beautiful ideas and methods, as well as surprising results.