We discuss hyperboloidal solutions to the constraint equations in the constant-mean-curvature setting, focusing attention on those solutions satisfying the shear-free condition. This condition is known to be necessary for a spacetime development of the data to admit a regular conformal boundary at future null infinity. We review results showing that such data can be constructed using the conformal method. We then present two results indicating that a wide variety of physical phenomena can be modeled within this class of data. These are (i) a boundary gluing result suitable for studying many-body problems and (ii) a density result suitable or approximating `typical' solutions by those with smooth conformal structures.