Elliptic curves are a special kind of curve that can be given as solutions to equations of the form y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6, with an added point at infinity. I will begin with an introduction to basic definitions, followed by a discussion of their significance, plus what we know (and don't know) about the group structure on their Q- and F_p-points.