The ability to generate and manipulate broadband chiral terahertz waves is essential for applications in material imaging, terahertz sensing, and diagnosis. It can also open up new possibilities for nonlinear terahertz spectroscopy and coherent control of chiral molecules and magnetic materials. The existing methods, however, often suffer from low efficiency, narrow bandwidth, or poor flexibility. Here, we propose a novel type of laser-driven terahertz emitters, consisting of metasurface-patterned magnetic multilayer heterostructures, that can overcome the shortcomings of the conventional approaches. Such hybrid terahertz emitters combine the advantages of spintronic emitters for being ultrabroadband, efficient, and highly flexible, as well as those of metasurfaces for the powerful control capabilities over the polarization state of emitted terahertz waves on an ultracompact platform. Taking a stripe-patterned metasurface as an example, we demonstrate the efficient generation and manipulation of broadband chiral terahertz waves. The ellipticity can reach >0.75 over a broad terahertz bandwidth (1 to 5 THz), representing a high-quality and efficient source for few-cycle circularly polarized terahertz pulses with stable carrier waveforms. Flexible control of ellipticity and helicity is also demonstrated with our systematic experiments and numerical simulations. We show that the terahertz polarization state is dictated by the interplay between laser-induced spintronic-origin currents and the screening charges/currents in the metasurfaces, which exhibits tailored anisotropic properties due to the predesigned geometric confinement effects. Our work opens a new pathway to metasurface-tailored spintronic emitters for efficient vector-control of electromagnetic waves in the terahertz regime.