  We discuss the definability of finite graphs in first-order logic with two
relation symbols for adjacency and equality of vertices. The logical depth
$D(G)$ of a graph $G$ is equal to the minimum quantifier depth of a sentence
defining $G$ up to isomorphism. The logical width $W(G)$ is the minimum number
of variables occurring in such a sentence. The logical length $L(G)$ is the
length of a shortest defining sentence. We survey known estimates for these
graph parameters and discuss their relations to other topics (such as the
efficiency of the Weisfeiler-Lehman algorithm in isomorphism testing, the
evolution of a random graph, quantitative characteristics of the zero-one law,
or the contribution of Frank Ramsey to the research on Hilbert's
Entscheidungsproblem). Also, we trace the behavior of the descriptive
complexity of a graph as the logic becomes more restrictive (for example, only
definitions with a bounded number of variables or quantifier alternations are
allowed) or more expressible (after powering with counting quantifiers).
