For once, the French movie poster pretty much nails it : A romantic comedy with zombies is exactly what this is.
The fun - yes, indeed! fun there is! - lies neither in the comedy (highly conventional) nor in the zombies (the post 2004 zombie onslaught in entertainment has killed the market by now) but in the combination of the two, in the irresistible dialogues and in the contagious pleasure Wright, Pegg and Frost evidently derives from working together.
'Shaun of the Dead' is their first collaboration but contains themes and motifs that have grown recurrent in their work (the pub as a central meeting place, the grisly horror scenes, the emphasis on friendship... Though no aliens in this one!).
You also want to see : 'Paul' (still my favourite!), 'Hot Fuzz' and, to complete your collection, 'The World's End'.