We describe the recent emergence of attosecond science, assessing the present state of the art and discussing several recent examples where attosecond electron dynamics has been studied in atomic and molecular systems. After introducing the generation and characterization of attosecond laser pulses, we describe the use of isolated attosecond pulses in a pump-probe experiment revealing the subcycle time dependence of a multiphoton ionization process and an experiment using the interference from a train of attosecond pulses to extract amplitude and phase information for electronic wave functions. We furthermore discuss experiments where ultrashort laser pulses with a reproducible waveform to control electron dynamics in the D2+ molecular ion on attosecond timescales. Attosecond science is coming of age and presently is reaching a level of maturity and sophistication that allows detailed investigations of the role of multielectron dynamics in physics and chemistry.