This paper calculates the dynamical polarization of graphene within the random phase approximation (RPA) for arbitrary wavevector, frequency, and doping. The authors derive expressions for the polarization in the long-wavelength limit ($q \to 0$) and the static case ($\omega = 0$). They find that at finite doping, the static susceptibility saturates to a constant value for low momenta and exhibits a discontinuity only in the second derivative at $q = 2k_F$. This results in Friedel oscillations in the presence of charged impurities, which decay with the same power law as the Thomas-Fermi contribution. The spin density oscillations in the presence of a magnetic impurity are also calculated. The dynamical polarization is used to determine the dispersion relation and decay rate of plasmons and acoustic phonons as a function of doping. The low screening of graphene, combined with the absence of a gap, leads to a significant stiffening of the longitudinal acoustic lattice vibrations. The paper discusses the implications of these findings for the transport and Raman properties of doped graphene.This paper calculates the dynamical polarization of graphene within the random phase approximation (RPA) for arbitrary wavevector, frequency, and doping. The authors derive expressions for the polarization in the long-wavelength limit ($q \to 0$) and the static case ($\omega = 0$). They find that at finite doping, the static susceptibility saturates to a constant value for low momenta and exhibits a discontinuity only in the second derivative at $q = 2k_F$. This results in Friedel oscillations in the presence of charged impurities, which decay with the same power law as the Thomas-Fermi contribution. The spin density oscillations in the presence of a magnetic impurity are also calculated. The dynamical polarization is used to determine the dispersion relation and decay rate of plasmons and acoustic phonons as a function of doping. The low screening of graphene, combined with the absence of a gap, leads to a significant stiffening of the longitudinal acoustic lattice vibrations. The paper discusses the implications of these findings for the transport and Raman properties of doped graphene.