  We study the classical H\'enon family $f_{a,b}:(x,y)\mapsto(1-ax^2+y,bx)$,
$0<a<2$, $0<b<1$, and prove that given an integer $k\geq 1$, there is a set of
parameters $E_k$ of positive two-dimensional Lebesgue measure so that
$f_{a,b}$, for $(a,b)\in E_k$, has at least $k$ attractive periodic orbits and
one strange attractor. A corresponding statement also holds for the
H\'enon-like families. The final main result of the paper is the existence,
within the classical H\'enon family, of a positive Lebesgue measure set of
parameters whose corresponding maps have two coexisting strange attractors.
