  In this article we consider exceptional sequences of invertible sheaves on
smooth complete rational surfaces. We show that to every such sequence one can
associate a smooth complete toric surface in a canonical way. We use this
structural result to prove various theorems on exceptional and strongly
exceptional sequences of invertible sheaves on rational surfaces. We construct
full strongly exceptional sequences for a large class of rational surfaces. For
the case of toric surfaces we give a complete classification of full strongly
exceptional sequences of invertible sheaves.
