In this study quantum chaos is discussed using the kicked harmonic oscillator as a model system. The kicked harmonic oscillator is characterized by an exceptional scenario of weak chaos: In the case of resonance between the frequency of the harmonic oscillator and the frequency of the periodic forcing, stochastic webs in phasespace are generated by the classical dynamics. For the quantum dynamics of this system it is shown, both numerically and analytically, that the resulting Husimi distributions in quantum phase space exhibit the same web-like structures as the classical webs. The quantum dynamics is characterized by diffusive energy growth - justas the classical dynamics in the channels of the webs. In the case of nonresonance, the classically diffusive dynamics is quantum mechanically suppressed. This bounded energy growth, which corresponds to localization in quantum phase space, is explained analytically by mapping the system onto the Anderson model. In this way, within the context of quantum chaos, the kicked harmonic oscillator is characterized by exhibiting its noteworthy geometrical and dynamical properties both classically and quantum mechanically, while at the same time there are also very distinct quantum deviations from classical properties, the most prominentexample being quantum localization.