We theoretically study time-resolved photoemission in atoms as probed by attosecond streaking. We review recent advances in the study of the photoelectric efect in the time domain and show that the experimentally accessible time shifts can be decomposed into distinct contributions that stem from the feld-free photoionization process itself and from probe-field induced corrections. We perform accurate quantum-mechanical as well as classical simulations of attosecond streaking for efective one-electron systems and determine all relevant contributions to the time delay with attosecond precision. In particular, we investigate the properties and limitations of attosecond streaking for the transition from short-ranged potentials (photodetachment) to long-ranged Coulomb potentials (photoionization). As an example for a more complex system, we study time-resolved photoionization for endohedral fullerenes A@C60 and discuss how streaking time shifts are modifed due to the interaction of the C60 cage with the probing infrared streaking field.