Special relativity is at the foundation of any physical theory. Though, its consequences leave us with unexpected results in kinematics, dynamics and, to a lesser extent, in electrodynamics. We introduce the theory of relativity including geometry and chronology, from the view of a local observer who encounters aberration and Doppler shift. The description of motion in Minkowski spacetime requires a careful distinction between different operational concepts of measuring time and space intervals, leading e.g. to the distinction of different velocities for the same process. The consequences in Newtonian dynamics, such as the obsolete relativistic mass, or in Maxwellian electrodynamics, the emergence of magnetic fields, are discussed in the light of experimental evidence. The final chapter discusses predictions for relativistic space flight. The material in "Radical Relativity" is a careful exposition of all the concepts and phrases that change with different paradigms. It is an unusual approach to spacetime that opens up new insights and deeper understanding.