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In these notes we investigate the cone of nef curves of projective varieties,which is the dual cone to the cone of pseudo-effective divisors. We prove astructure theorem for the cone of nef curves of projective $\mathbbQ$-factorial klt pairs of arbitrary dimension from the point of view of theMinimal Model Program. This is a generalization of Batyrev's structure theoremfor the cone of nef curves of projective terminal threefolds.Comment: 15 pages. v2: Completely rewritten paper. Structure theorem for the  cone of nef curves proved in arbitrary dimension using results of Birkar,  Cascini, Hacon and McKernan. To appear in Mathematische Zeitschrif

The cone of pseudo-effective divisors of log varieties after Batyrev