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In 1878 \'E. Lucas proved a remarkable result which provides a simple way tocompute the binomial coefficient ${n\choose m}$ modulo a prime $p$ in terms ofthe binomial coefficients of the base-$p$ digits of $n$ and $m$: {\it If $p$ isa 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 ofLucas}), plays an important role in Number Theory and Combinatorics. In thisarticle, consisting of six sections, we provide a historical survey of Lucastype congruences, generalizations of Lucas' theorem modulo prime powers, Lucaslike theorems for some generalized binomial coefficients, and some theirapplications.  In Section 1 we present the fundamental congruences modulo a prime includingthe famous Lucas' theorem. In Section 2 we mention several known proofs andsome consequences of Lucas' theorem. In Section 3 we present a number ofextensions and variations of Lucas' theorem modulo prime powers. In Section 4we consider the notions of the Lucas property and the double Lucas property,where we also present numerous integer sequences satisfying one of theseproperties or a certain Lucas type congruence. In Section 5 we collect severalknown Lucas type congruences for some generalized binomial coefficients. Inparticular, this concerns the Fibonomial coefficients, the Lucas $u$-nomialcoefficients, the Gaussian $q$-nomial coefficients and their generalizations.Finally, some applications of Lucas' theorem in Number Theory and Combinatoricsare given in Section 6.

Lucas' theorem: its generalizations, extensions and applications (1878--2014)