Polynomial equations and their solutions form a cornerstone of mathematics. Solutions with rational coordinates are particularly intriguing; a fantastic surprise is the great difficulty of determining the mere existence of a rational solution to a given equation (let alone the complete set). We will discuss this problem in two cases, diagonal cubic surfaces, \[ax^3+by^3+cz^3+d=0,\] and degree 2 del Pezzo surfaces, \[ax^4+by^4+cx^2y^2+d=z^2.\] A surprising and successful modern approach, the Brauer--Manin obstruction, employs tools from linear algebra, geometry and non-commutative algebra. I will discuss a collection of interesting and motivating examples with simultaneous historical and modern interest, and also explain some of the tools and techniques that form the backbone of my research program.