  In this paper we present a simple unifying approach to prove several
statements about intersecting and cross-intersecting families, including the
Erd\H os--Ko--Rado theorem, the Hilton--Milner theorem, a theorem due to Frankl
concerning the size of intersecting families with bounded maximal degree, and
versions of results on the sum of sizes of non-empty cross-intersecting
families due to Frankl and Tokushige. Several new stronger results are also
obtained.
  Our approach is based on the use of regular bipartite graphs. These graphs
are quite often used in Extremal Set Theory problems, however, the approach we
develop proves to be particularly fruitful.
