One of the important topics in analytic number theory is the distribution of prime numbers with various arithmetic constraints, which is just a fancy way of saying "How many primes of my favorite type are there?" Some examples of types of primes that we are interested in include: primes that are one less than (alternatively, one greater than) a multiple of 4; primes that end in the digit 1 (or 3 or 7 or 9); primes that differ (or don't differ) from the square of an integer by a multiple of 13. When two of these types of primes are sufficiently closely related, a race between them develops: "Is there more of the first type of prime than the second?" Like many questions, this one might have the answer "yes", "no", or "sometimes". Also like many questions, this one has several different reformulations that are mathematically precise. In this colloquium (which will be accessible to anyone who knows what a prime is), we will describe these mathematical variants of the race question and show that the answer is "yes" and "no" and "sometimes" all at once. The "sometimes" answer will be particularly interesting.