A number of articles in these proceedings will discuss the affine sieve method which in general produces some (usually unspecified) number of almost-primes in an orbit of affine linear maps [Bourgain et al. 2010a; Salehi Golsefidy and Sarnak 2011]. Our goal in these notes is to highlight several occasions in which actual primes can be produced in the thin setting.1 In the results surveyed below, these come from the stronger statement of an “almost every” local-to-global phenomenon, which has recently been established in several a priori unrelated settings by Bourgain and the author. The key idea is to replace sieving with the circle method, and develop the latter in the setting of thin orbits. We now state these results.