  This survey is a presentation of the arguments in the proof that Henon-like
maps f_a(x,y)=(1-a x^2,0) + R(a,x,y) with |R(a,x,y)|< b have a "strange
attractor", with positive Lebesgue probability in the parameter "a", if the
perturbation size "b" is small enough. We first sketch a "geometric model" of
the strange attractor in this context, emphasising some of its key geometrical
properties, and then focus on the construction and estimates required to show
that this geometric model does indeed occur for many parameter values. Our
ambitious aim is to provide an exposition at one and the same time intuitive,
synthetic, and rigorous. We think of this text as an introduction and study
guide to the original papers in which the results were first proved. We shall
concentrate on describing in detail the overall structure of the argument and
the way it breaks down into its (numerous) constituent sub-arguments, while
referring the reader to the original sources for detailed technical arguments.
