  Serrrano's Conjecture says that if $L$ is a strictly nef line bundle on a
smooth projective variety $X$, then $K_X+tL$ is ample for $ t > dim X+1$. In
this paper I will prove a few cases of this conjecture. I will also prove a
generalized version of this conjecture (due to Campana, Chen and Peternell) for
surfaces. In the last section, assuming the SHGH conjecture, I will give a
series of examples of strictly nef non ample divisors on surfaces of arbitary
Kodaira dimension.
