A wave packet theory of scatterings is presented in which time delay (or advance) phenomena are discussed. We start from the wave packet: (G, α, g and β being real functions of the energy E). g exp (i β) is the scattering amplitude. For elastic scattering it is shown that the observable delay time for incident beams with nearly monochromatic energy E is just half of Wigner's definition of delay time and F. T. Smith's collision lifetime, if the phase α(E) of the incident wave packet is a linear function of energy. However, this assumption is largely violated under certain experimental conditions; hence the delay time is not always given in terms of the scattering matrix elements only. This fact limits the hope that the S-matrix alone can give a space-time description of collision events. The general formula of delay time < Δt> has the following form: The observable scattering cross section <σ(θ) > is given by , which is independent of α, unlike the delay time. The spreading of the wave packet in space or time before and after the scattering is considered explicitly. Examples of scattered wave packet are given for resonance scattering with background (so-called potential scattering) phase shift.