| Summary: | We show that a decorated knot concordance C from K to K' induces a homomorphism F_C on knot Floer homology that preserves the Alexander and Maslov gradings. Furthermore, it induces a morphism of the spectral sequences to HF^(S^3) = Z_2 that agrees with F_C on the E^1 page and is the identity on the E^infinity page. It follows that F_C is non-vanishing on HFK^_0(K, \tau(K)). We also obtain an invariant of slice disks in homology 4-balls bounding S^3.If C is invertible, then F_C is injective, hence dim HFKh_j(K,i) <= dim HFKh_j(K',i) for every i, j in Z. This implies an unpublished result of Ruberman that if there is an invertible concordance from the knot K to K', then g(K) <= g(K'), where g denotes the Seifert genus. Furthermore, if g(K) = g(K') and K' is fibred, then so is K.
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