Efficiency and power are the two main parameters to evaluate the performance of heat engines. The Carnot efficiency is the upper limit in the efficiency of a reversible heat engine. Due to the requirement of infinite operation time in quasi-static processes, the output power of a reversible heat engine approaches zero, with no practical value. Therefore, how to improve the efficiency of a heat engine while maintaining its power for practical purposes is an important scientific challenge in thermodynamics. Finite time thermodynamics, born in the first half of the last century, is developing rapidly to provide the necessary scientific support for this challenge. This paper sketches the early development and present status of finite-time thermodynamics, focusing on current investigations on the power—efficiency trade-off relation for finite-time heat engines. To explore the relation between this trade-off and the irreversibility of finite-time thermodynamic cycles, we introduce our recent theoretical and experimental studies on the irreversible entropy generation in finite-time isothermal processes. Possible future trends and applications of finite-time thermodynamics and the non-equilibrium physics of finite systems are also previewed.