  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 $\widehat{HF}(S^3) \cong \mathbb{Z}_2$ that agrees with $F_C$ on the $E^1$
page and is the identity on the $E^\infty$ page. It follows that $F_C$ is
non-vanishing on $\widehat{HFK}_0(K, \tau(K))$. 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 $\dim \widehat{HFK}_j(K,i) \le \dim
\widehat{HFK}_j(K',i)$ for every $i$, $j \in \mathbb{Z}$. This implies an
unpublished result of Ruberman that if there is an invertible concordance from
the knot $K$ to $K'$, then $g(K) \le g(K')$, where $g$ denotes the Seifert
genus. Furthermore, if $g(K) = g(K')$ and $K'$ is fibred, then so is $K$.
