  The Gruenberg-Kegel graph $\Gamma(G)$ associated with a finite group $G$ has
as vertices the prime divisors of $|G|$, with an edge from $p$ to $q$ if and
only if $G$ contains an element of order $pq$. This graph has been the subject
of much recent interest; one of our goals here is to give a survey of some of
this material, relating to groups with the same Gruenberg-Kegel graph. However,
our main aim is to prove several new results. Among them are the following.
  - There are infinitely many finite groups with the same Gruenberg-Kegel graph
as the Gruenberg-Kegel of a finite group $G$ if and only if there is a finite
group $H$ with non-trivial solvable radical such that $\Gamma(G)=\Gamma(H)$.
  - There is a function $F$ on the natural numbers with the property that if a
finite $n$-vertex graph whose vertices are labelled by pairwise distinct primes
is the Gruenberg-Kegel graph of more than $F(n)$ finite groups, then it is the
Gruenberg-Kegel graph of infinitely many finite groups. (The function we give
satisfies $F(n)=O(n^7)$, but this is probably not best possible.)
  - If a finite graph $\Gamma$ whose vertices are labelled by pairwise distinct
primes is the Gruenberg-Kegel graph of only finitely many finite groups, then
all such groups are almost simple; moreover, $\Gamma$ has at least three
pairwise non-adjacent vertices, and $2$ is non-adjacent to at least one odd
vertex.
  - Groups whose power graphs, or commuting graphs, are isomorphic have the
same Gruenberg-Kegel graph.
  - The groups ${^2}G_2(27)$ and $E_8(2)$ are uniquely determined by the
isomorphism types of their Gruenberg-Kegel graphs.
  In addition, we consider groups whose Gruenberg-Kegel graph has no edges.
These are the groups in which every element has prime power order, and have
been studied under the name \emph{EPPO groups}; completing this line of
research, we give a complete list of such groups.
