Recent advances in the generation of well characterized sub-femtosecond laser pulses have opened up unpredicted opportunities for the real-time observation of ultrafast electronic dynamics in matter. Such attosecond chronoscopy allows a novel look at a wide range of fundamental photophysical and photochemical processes in the time domain, including Auger and autoionization processes, photoemission from atoms, molecules, and surfaces, complementing conventional energy-domain spectroscopy. Attosecond chronoscopy raises fundamental conceptual and theoretical questions as which novel information becomes accessible and which dynamical processes can be controlled and steered. These questions are currently a matter of lively debate which we address in this review. We will focus on one prototypical case, the chronoscopy of the photoelectric effect by attosecond streaking. Is photoionization instantaneous or is there a finite response time of the electronic wavefunction to the photoabsorption event? Answers to this question turn out to be far more complex and multi-faceted than initially thought. They touch upon fundamental issues of time and time delay as observables in quantum theory. We review recent progress of our understanding of time-resolved photoemission from atoms, molecules, and solids. We will highlight the unresolved and open questions and we point to future directions aiming at the observation and control of electronic motion in more complex nanoscale structures and in condensed matter.