Since the work of Dirichlet, it has been known that there are infinitely many primes in an arithmetic progression a (mod q) when gcd(a, q) = 1. One may then ask: when does the least such prime p occur? In his celebrated result from 1944, Linnik famously showed that p is bounded by O(qL) for some absolute constant L > 0 known as ”Linnik’s constant”. Since then many authors have computed admissible values of Linnik’s constant with the current world record being L = 5.2 by Xylouris (2011) refining work of Heath-Brown (1995). We will investigate the proof of Linnik’s theorem and describe work related to the computation of Linnik’s constant. Time permitting, we will briefly discuss some generalizations of Linnik’s theorem.