The concept of one-parameter operator semigroups is one of the oldest and most well studied parts of the theory of operators in Banach spaces. It was developed in the middle of the last century as a natural approach to investigation of PDEs and has proved to have many useful and important applications. It is difficult to observe this concept in its entirety, in the sense that there are dozens of textbooks and thousands of research papers directly related to one-parameter operator semigroups. We do not attempt such an overview in this book but devote our attention to some relatively new topics from this theory. Namely, we present and discuss some non-spectral methods that have been developed over the last two decades for the investigation of asymptotic behavior of operator semigroups.