Arithmetic dynamics studies arithmetic questions arising from the iteration of maps.

In this expository talk, I will introduce the subject by surveying some known results for the family

\[f_c(x)=x^2+c,\qquad c\in\mathbb{Q}.\]

The guiding question will be: what finite rational orbit structures can occur for these maps?

I will begin with periodic and preperiodic points, and explain how they define directed graphs whose isomorphism classes are called portraits. Along the way, I will discuss Northcott's theorem, the Uniform Boundedness Conjecture, and the connection with rational points on algebraic curves.