  Introduced by Erd\H{o}s in 1950, a covering system of the integers is a
finite collection of arithmetic progressions whose union is the set
$\mathbb{Z}$. Many beautiful questions and conjectures about covering systems
have been posed over the past several decades, but until recently little was
known about their properties. Most famously, the so-called minimum modulus
problem of Erd\H{o}s was resolved in 2015 by Hough, who proved that in every
covering system with distinct moduli, the minimum modulus is at most $10^{16}$.
  In this paper we answer another question of Erd\H{o}s, asked in 1952, on the
number of minimal covering systems. More precisely, we show that the number of
minimal covering systems with exactly $n$ elements is \[ \exp\left(
\left(\frac{4\sqrt{\tau}}{3} + o(1)\right) \frac{n^{3/2}}{(\log n)^{1/2}}
\right) \] as $n \to \infty$, where \[ \tau = \sum_{t = 1}^\infty \left( \log
\frac{t+1}{t} \right)^2. \] En route to this counting result, we obtain a
structural description of all covering systems that are close to optimal in an
appropriate sense.
