The eccentricitye(v) of a vertexv of a connected graphG is the maximum distance fromv among the vertices ofG. A nondecreasing sequencea1,a2, ...,ap of nonnegative integers is said to be an eccentric sequence if there exists a connected graphG of orderp whose vertices can be labelledv1,v2, ...,vp so thate(vi)=ai for alli. Several properties of eccentric sequences are exhibited, and a necessary and sufficient condition for a sequence to be eccentric is presented. Sequences which are the eccentricity sequences of trees are characterized. Some properties of the eccentricity sequences of self-complementary graphs are obtained. It is shown that the radius of a nontrivial self-complementary graph is two.