  In 1878 \'E. Lucas proved a remarkable result which provides a simple way to
compute the binomial coefficient ${n\choose m}$ modulo a prime $p$ in terms of
the binomial coefficients of the base-$p$ digits of $n$ and $m$: {\it If $p$ is
a prime, $n=n_0+n_1p+\cdots +n_sp^s$ and $m=m_0+m_1p+\cdots +m_sp^s$ are the
$p$-adic expansions of nonnegative integers $n$ and $m$, then
  \begin{equation*} {n\choose m}\equiv \prod_{i=0}^{s}{n_i\choose m_i}\pmod{p}.
  \end{equation*}}
  The above congruence, the so-called {\it Lucas' theorem} (or {\it Theorem of
Lucas}), plays an important role in Number Theory and Combinatorics. In this
article, consisting of six sections, we provide a historical survey of Lucas
type congruences, generalizations of Lucas' theorem modulo prime powers, Lucas
like theorems for some generalized binomial coefficients, and some their
applications.
  In Section 1 we present the fundamental congruences modulo a prime including
the famous Lucas' theorem. In Section 2 we mention several known proofs and
some consequences of Lucas' theorem. In Section 3 we present a number of
extensions and variations of Lucas' theorem modulo prime powers. In Section 4
we consider the notions of the Lucas property and the double Lucas property,
where we also present numerous integer sequences satisfying one of these
properties or a certain Lucas type congruence. In Section 5 we collect several
known Lucas type congruences for some generalized binomial coefficients. In
particular, this concerns the Fibonomial coefficients, the Lucas $u$-nomial
coefficients, the Gaussian $q$-nomial coefficients and their generalizations.
Finally, some applications of Lucas' theorem in Number Theory and Combinatorics
are given in Section 6.
