Swann grew up on a farm in southern Pennsylvania, before attending Phillips Academy and then Columbia College, where she studied Comparative Literature (French and German) and creative writing with Mary Gordon, graduating in 1994.

She pursued her graduate studies at the Sorbonne, Université de Paris VII, earning her master's degree in 1997 with a thesis on the style of Marcel Proust.

Swann's work first appeared in Ploughshares, in an issue guest-edited by Mary Gordon, and she won a Cohen Award for that short story, "Flower Children", in 1997.

In her writing class at Columbia, Mary Gordon, taking my efforts seriously, pressed me further towards it, also introducing me to the Austrian writer Ingeborg Bachmann, whose work eventually led me to find the form in which to say what I wanted to say."

Her March 2013 story for The New York Times Magazine, "The Professor, the Bikini Model and the Suitcase Full of Trouble" was optioned for a feature film by Fox Searchlight.