We present first-principles approaches based on density functional theory for calculating positron states and annihilation characteristics in condensed matter. The treatment of the electron-positron correlation effects (the enhancement of the electron density at the positron with respect to mean-field density) is shown to play a crucial role when calculating the annihilation rates. A generalized gradient approximation (GGA) takes the strong inhomogeneities of the electron density in the ion core region into account and reproduces well the experimental total annihilation rates (inverses of the positron lifetimes) by suppressing the rates given by a local density approximation (LDA). The GGA combined with an electron-state-dependent enhancement scheme gives a good description for the momentum distributions of the annihilating positron-electron pairs reproducing accurately the trends observed in the angular correlation (ACAR) or Doppler broadening measurements of the annihilation radiation. The combination of the present positron lifetime and momentum density calculations with the corresponding measurements yields a unique tool for defect identification. Especially, the investigation of various vacancy-type defects in semiconductors able to trap positrons will be an important field for these methods. We will show that the identification of vacancy-impurity complexes in highly n-Type Si and the study of the SiO$_2$/Si interface are particularly interesting applications.