  We summarize results on the Penrose inequality bounding the ADM-mass or the
Bondi mass in terms of the area of an outermost apparent horizon for
asymptotically flat initial data of Einstein's equations. We first recall the
proof, due to Geroch and to Jang and Wald, of monotonicity of the
Geroch-Hawking mass under a smooth inverse mean curvature flow for data with
non-negative Ricci scalar, which leads to a Penrose inequality if the apparent
horizon is a minimal surface.We then sketch a proof of the Penrose inequality
of Malec, Mars and Simon which holds for general horizons and for data
satisfying the dominant energy condition, but imposes (in addition to smooth
inverse mean curvature flow) suitable restrictions on the data on a spacelike
surface. These conditions can, however, at least locally be fulfilled by a
suitable choice of the initial surface in a given spacetime. Remarkably, they
are also (formally) identical to ones employed earlier by Hayward in order to
define a 2+1 foliation on null surfaces, with respect to which the Hawking mass
is again monotonic.
