We first recall the main features of Mach's radical attitude to the "philosophy of science", illustrated best by his criticism of absolute space and time, and discuss the impact of Mach's ideas on the birth of special and general relativity. The experiment to demonstrate "Machian effects" of dragging of inertial frames by a rotating body (carried out by four gyroscopes placed in an orbit around the Earth) was conceived in 1959 but only in 2014 was the satellite launched. The last results from the experiment were finally summarised at the end of 2015: they confirm the dragging as predicted by general relativity with an accuracy of 19 %. We show that a rotating black hole can even drag magnetic field lines and indicate possible astrophysical meaning of processes of this type. in the concluding parts, our investigation of dragging effects by gravitational waves and the study of Mach's principle in the context of cosmology, obtained in collaboration with Donald Lynden-Bell (Cambridge) and Joseph Katz (Jerusalem), are briefly reviewed., Jiří Bičák., and Obsahuje bibliografii