Juhasz, Zemke and Thurston established that the hat version of Heegaard Floer homology fits into the framework of a TQFT. The notion of a sutured manifold plays an important role in establishing functoriality. In this talk I'll outline a framework for understanding HF^hat as an extended TQFT in dimensions 2-3-4, with the 2-3 part given by the bordered Floer homology of Lipshitz-Ozsvath-Thurston and Zarev. Again, sutures play an essential role.
