  This is an intuitive survey of extrinsic and intrinsic notions of convergence
of manifolds complete with pictures of key examples and a discussion of the
properties associated with each notion. We begin with a description of three
extrinsic notions which have been applied to study sequences of submanifolds in
Euclidean space: Hausdorff convergence of sets, flat convergence of integral
currents, and weak convergence of varifolds. We next describe a variety of
intrinsic notions of convergence which have been applied to study sequences of
compact Riemannian manifolds: Gromov-Hausdorff convergence of metric spaces,
convergence of metric measure spaces, Instrinsic Flat convergence of integral
current spaces, and ultralimits of metric spaces. We close with a speculative
section addressing possible notions of intrinsic varifold convergence,
convergence of Lorentzian manifolds and area convergence.
