This talk is a survey of “prime number races”. Around 1850, Chebyshev noticed that for any given value of $x$, there always seem to be more primes of the form $4n+3$ less than $x$ than there are of the form $4n+1$. Similar observations have been made with primes of the form $3n+2$ and $3n+1$, primes of the form $10n+3,10n+7$ and $10n+1,10n+9$, and many others besides. More generally, one can consider primes of the form $qn+1,qn+bn,qn+c,\dots$ for our favorite constants $q,a,b,c,\dots$ and try to figure out which forms are “preferred” over the others $-$ not to mention figuring out what, precisely, being “preferred” means. All of these “races” are related to the function $\pi(x)$ that counts the number of primes up to $x$, which has both an asymptotic formula with a wonderful proof and an associated “race” of its own; and the attempts to analyze these races are closely related to the Riemann hypothesis $-$ the most famous open problem in mathematics.
We describe these phenomena, in an accessible way, in greater detail; we provide examples of computations that demonstrate the “preferences” described above; and we explain the efforts that have been made at understanding the underlying mathematics.