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

Strictly nef divisors and some remarks on a conjecture of Serrano