The past two decades have witnessed an ever-growing number of emerging applications that utilize terahertz (THz) waves, ranging from advanced biomedical imaging, through novel security applications, fast wireless communications, and new abilities to study and control matter in all of its phases. The development and deployment of these emerging technologies is however held back, due to a substantial lack of simple methods for efficient generation, detection and manipulation of THz waves. Recently it was shown that uniform nonlinear metasurfaces can efficiently generate broadband single-cycle THz pulses. Here we show that judicious engineering of the single-emitters that comprise the metasurface, enables to obtain unprecedented control of the spatiotemporal properties of the emitted THz wavepackets. We specifically demonstrate generation of propagating spatiotemporal quadrupole and few-cycles THz pulses with engineered angular dispersion. Our results place nonlinear metasurfaces as a new promising tool for generating application-tailored THz fields with controlled spatial and temporal characteristics.