The Brauer-Manin obstruction and the finite descent obstruction are ways to prove that a variety over a global field has no rational points, by a 'local-global' comparison with adelic points. I will talk about some recent general results of myself and Schlank concerning the Brauer-Manin obstruction, showing that the Brauer-Manin obstruction always detects existence of rational points if one restricts to small enough Zariski open coverings. I will then talk about how this relates to the etale homotopy obstruction of Harpaz and Schlank, which we used to prove a result about the (etale) Brauer-Manin obstruction in fibrations. Finally, I will mention ongoing work in attempting to apply the etale homotopy obstruction to a del Pezzo surface over Q_p(t).