  We construct $d$-closed and $dd^c$-closed positive currents associated to a
holomorphic map $\phi$ via cluster points of normalized weighted truncated
image currents. They are constructed using analogues of the Ahlfors length-area
inequality in higher dimensions. Such classes of currents are referred to as
Ahlfors currents. We give some applications to equidistribution problems in
value distribution theory.
