This paper presents a detailed calculation of the dynamical polarization of graphene within the random phase approximation (RPA) for arbitrary wavevector, frequency, and doping. The results show that at finite doping, the static susceptibility saturates to a constant value for low momenta. At q = 2k_F, the static susceptibility has a discontinuity only in the second derivative. The presence of a charged impurity leads to Friedel oscillations, which decay with the same power law as the Thomas-Fermi contribution, which is always dominant. The spin density oscillations in the presence of a magnetic impurity are also calculated. The dynamical polarization is used to calculate 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 also discusses the static screening of graphene, showing that the induced charge density has two contributions: a non-oscillating part from the long-wavelength behavior of the polarization and an oscillatory part from the non-analyticity of the polarization at ħv_Fq = 2μ. The oscillatory part decays as cos(2k_Fr)/r^3, while the non-oscillating part decays as 1/r^3. The induced spin density also decays as 1/r^3. The plasmon dispersion is determined by solving for ε(q, ω_p - iγ) = 0, where γ is the decay rate of the plasmons. The plasmon dispersion shows a √q behavior in the long wavelength regime. The acoustic phonon dispersion is strongly dependent on the chemical potential, and the sound velocity approaches the Fermi velocity at low doping. The paper concludes that the dynamical polarization of graphene has important implications for the electronic and optical properties of doped graphene.This paper presents a detailed calculation of the dynamical polarization of graphene within the random phase approximation (RPA) for arbitrary wavevector, frequency, and doping. The results show that at finite doping, the static susceptibility saturates to a constant value for low momenta. At q = 2k_F, the static susceptibility has a discontinuity only in the second derivative. The presence of a charged impurity leads to Friedel oscillations, which decay with the same power law as the Thomas-Fermi contribution, which is always dominant. The spin density oscillations in the presence of a magnetic impurity are also calculated. The dynamical polarization is used to calculate 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 also discusses the static screening of graphene, showing that the induced charge density has two contributions: a non-oscillating part from the long-wavelength behavior of the polarization and an oscillatory part from the non-analyticity of the polarization at ħv_Fq = 2μ. The oscillatory part decays as cos(2k_Fr)/r^3, while the non-oscillating part decays as 1/r^3. The induced spin density also decays as 1/r^3. The plasmon dispersion is determined by solving for ε(q, ω_p - iγ) = 0, where γ is the decay rate of the plasmons. The plasmon dispersion shows a √q behavior in the long wavelength regime. The acoustic phonon dispersion is strongly dependent on the chemical potential, and the sound velocity approaches the Fermi velocity at low doping. The paper concludes that the dynamical polarization of graphene has important implications for the electronic and optical properties of doped graphene.