%\section{Introduction}\label{sec1:introduction}

Distance-regular graphs are graphs with a lot of combinatorial
symmetry, in the sense that given an arbitrary ordered pair of
vertices at distance $h$, the number of vertices that are at
distance $i$ from the first vertex and distance $j$ from the second
is a constant (i.e., does not depend on the chosen pair) that only
depends on $h,i$, and $j$. Biggs introduced distance-regular
graphs, by observing that several combinatorial and linear
algebraic properties of distance-transitive graphs were holding for
this wider class of graphs, see Biggs' monograph \cite{biggs} from
1974. Well-known examples are the Hamming graphs and the Johnson
graphs, as these graphs link the subject of distance-regular graphs
to coding theory and design theory, respectively. But there are
many more interesting links to other subjects, such as finite group
theory (and distance-transitive graphs), representation theory,
finite geometry, association schemes, and orthogonal polynomials.
Moreover, distance-regular graphs are frequently used as test
instances for problems on general graphs and other combinatorial
structures, such as problems related to random walks and from
combinatorial optimization. An example is Hoffman's (unpublished;
see \cite[Thm.~3.5.2]{BrHa}) coclique bound, which was first proved
by Delsarte \cite[p.~31]{del} for distance-regular graphs with
diameter two (also known as strongly regular graphs), as an example
of his linear programming method. Distance-regular graphs have
applications in several fields besides the already mentioned
classical coding and design theory, such as (quantum) information
theory, diffusion models, (parallel) networks, and even finance.

In this survey of distance-regular graphs, we give an overview of some
developments in the area of distance-regular graphs since the monograph `BCN'
by Brouwer, Cohen, and Neumaier \cite{bcn} from 1989 was written. This
influential monograph, which is almost like an encyclopedia of distance-regular
graphs, inspired many researchers to work on distance-regular graphs, such as
the authors of this survey. Since then, many papers have been written, many
more than the ones we will discuss in this overview. We intend to discuss the
most relevant developments of the past twenty-seven years, realizing that `most
relevant' is quite subjective. Perhaps we should say that we give our personal
view on the past twenty-seven years. The same is true when we discuss the major
open problems in the area. A recent major breakthrough is the proof of one of
the Bannai-Ito conjectures made in the influential monograph by Bannai and Ito \cite{bi}
from 1984, i.e., the one that states that there are finitely many
distance-regular graphs with given valency (at least three). Just as important
is the theorem stating that there are finitely many non-geometric
distance-regular graphs with both valency and diameter at least three and
smallest eigenvalue at least a given number; a generalization of a well-known
result about strongly regular graphs. The classification of tridiagonal pairs
is an example of an important recent breakthrough in algebraic combinatorics
that is completely inspired by the major (still) open problem of classifying
the $Q$-polynomial distance-regular graphs. The construction of the twisted
Grassmann graphs, that is, of this family of strange examples that were not
expected to be in the picture, gave a better perspective on how difficult this
classification problem really is. It seems to suggest that the problem cannot
be solved just by algebraic methods. In addition, we need to better understand
geometric distance-regular graphs.

This survey is organized as follows. After this brief introduction, we present
an introduction to distance-regular graphs for the reader that is unfamiliar
with the subject. We then present the classical examples of distance-regular
graphs, and an overview of the most important constructions since `BCN' \cite{bcn}. In
Section \ref{sec:morebackground}, we give more necessary and advanced
background for the remaining part of the paper. We then treat several subjects
in Sections \ref{sec:Qpol}-\ref{sec:mult}, for example $Q$-polynomial
distance-regular graphs, the Terwilliger algebra, the Bannai-Ito conjecture,
geometric distance-regular graphs, and spectral characterizations. In Section
\ref{sec:applications}, we discuss important applications of distance-regular
graphs, namely in combinatorial optimization and in the area of
random (classical and quantum) walks (which model diffusion models, dynamic
stock portfolios, and the abelian sandpile, for example). In Section
\ref{sec:misc}, we then discuss some miscellaneous topics, and in Section
\ref{sec:tables} we report progress on the `feasibility' and `uniqueness' of
the intersection arrays that were listed in the tables of parameter sets of
distance-regular graphs in `BCN' \cite{bcn}. We conclude with a section on open problems
and some directions for future research.

Note that we will focus our attention on distance-regular graphs with diameter at least three. We do not completely
exclude strongly regular graphs (the diameter two case), but we are of the opinion that they form a subject of their
own. A separate survey of strongly regular graphs would therefore be warmly welcomed. For some information we refer to
the recent book by Brouwer and Haemers on spectra of graphs \cite[Ch.~9]{BrHa} and the paper by Cohen and Pasechnik \cite{CPsrg}.
Also bipartite distance-regular graphs
with diameter three form a separate subject. These graphs are equivalent to symmetric designs, for which we refer to
the monograph by Ionin and Shrikhande \cite{symmetricdesigns}.
