The purpose of this thesis is to study link cobordisms. The main tool we use to do so is given by Juhász’s cobordism maps, although we sometimes make use of a recent construction by Zemke. Given a decorated link cobordism, that is, a link cobordism with additional structure, there is a map induced at the level of link Floer homology (HFL, also denoted HFK for knots).

When X is a decorated concordance from K_0 to K_1, we prove that the cobordism map F preserves the natural bigrading of HFK, and that there is a morphism of spectral sequences from HFK(K_i) to HF(S^3) which agrees with F on the first page, and with the identity of HF(S^3) on the infinity page. We use this result to obstruct the existence of invertible concordances between given knots, and to define a non-vanishing element of HFK(K) associated to a slice disc for the knot K.

We then give a full description of the maps induced by elementary decorated link cobordisms, which generate all decorated link cobordisms. In particular, we relate the map associated to a saddle cobordism to a skein exact triangle which generalises Ozsváth and Szabó’s oriented skein exact triangle in HFL to decorated links.

Lastly, we use this description to prove that the TQFT defined by HFL on a category of unlinks and cobordisms between them agrees with Khovanov’s reduced TQFT.