The étale topology on schemes is a testament to the influence of algebraic topology on algebraic geometry. Étale realizations of topological invariants, such as the fundamental group and singular cohomology, played a central role in the development of french-school algebraic geometry. This exposé will tell the story of the étale homotopy type of a scheme, which provides a general framework for realizing homotopy-theoretic constructions in étale algebraic geometry. As an application of this, I will present an obstruction theory for finding rational points on varieties. At the level of $\pi_1$ this theory will recover classical obstructions due to Grothendieck. I'll conclude with the question of exhibiting explicit examples in which the classical $\pi_1$-obstruction vanishes, but higher obstruction classes prevent the existence of rational points.