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I will discuss some of the driving questions in the area of  
Rational points on algebraic varieties (these can be thought of as  
solutions over fields of number-theoretic interest to systems of  
homogeneous polynomial equations).  I will focus on the case of smooth  
rational surfaces, and discuss some results concerning the arithmetic  
of del Pezzo surfaces of degree 1.  I will also explain how these  
results complete a qualitative picture of basic arithmetic phenomena  
among smooth rational surfaces.  Along the way I will go over concepts  
like weak approximation and the computation of Brauer-Manin  
obstructions; I will not assume previous knowledge of them.
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