A simple theory is presented to study the elementary processes of two-
photon photoemission (2PPE) from clean metals and adsorbates on metal
surfaces. The model can be applied to metals having the occupied surfa
ce state and unoccupied image-potential state and also to adsorbates w
hose unoccupied states are populated from the occupied valence levels
or the substrate by photoexcitation. The 2PPE spectrum is calculated o
n the basis of a simple energy diagram. It is demonstrated that one-ph
oton photoemission from an initially unoccupied state (a fixed interme
diate state which is populated by photon-driven electron excitation fr
om an occupied metal state), is accompanied by two-photon photoemissio
n from the occupied initial state via the unoccupied state as intermed
iate. When an electron is nonresonantly excited from narrow metal stat
es such as d-bands or surface states, the lineshape of the correspondi
ng 2PPE spectrum exhibits a characteristic asymmetry due to the underl
ying direct 2PPE from the occupied states. At resonant excitation from
the narrow initial state to the intermediate state the 2PPE spectra e
xhibit a narrowing of the linewidth. It is shown that the lifetime of
the unoccupied state cannot be deduced from the linewidth of the 2PPE
spectra if a width of the initial state is comparable or narrower than
the unoccupied state. No resonance enhancement and narrowing effect i
n the 2PPE spectra appear when a broad bulk continuum serves as an ini
tial state of the 2PPE process.