  We analyze the geometry of scaling limits of near-critical 2D percolation,
i.e., for $p=p_c+\lambda\delta^{1/\nu}$, with $\nu=4/3$, as the lattice spacing
$\delta \to 0$. Our proposed framework extends previous analyses for $p=p_c$,
based on $SLE_6$. It combines the continuum nonsimple loop process describing
the full scaling limit at criticality with a Poissonian process for marking
double (touching) points of that (critical) loop process. The double points are
exactly the continuum limits of "macroscopically pivotal" lattice sites and the
marked ones are those that actually change state as $\lambda$ varies. This
structure is rich enough to yield a one-parameter family of near-critical loop
processes and their associated connectivity probabilities as well as related
processes describing, e.g., the scaling limit of 2D minimal spanning trees.
