  We introduce a counterpart to the notion of vertex disjoint tilings by copy
of a fixed graph F to the setting of graphons. The case F=K_2 gives the notion
of matchings in graphons. We give a transference statement that allows us to
switch between the finite and limit notion, and derive several favorable
properties, including the LP-duality counterpart to the classical relation
between the fractional vertex covers and fractional matchings/tilings, and
discuss connections with property testing. As an application of our theory, we
determine the asymptotically almost sure F-tiling number of inhomogeneous
random graphs \mathbb{G}(n,W). As another application, in an accompanying paper
[Hladky, Hu, Piguet: Komlos's tiling theorem via graphon covers, preprint] we
give a proof of a strengthening of a theorem of Komlos [Komlos: Tiling Tur\'an
Theorems, Combinatorica, 2000].
