The transverse emittance of the electron beam is a fundamental parameter in linac-based x-ray free-electron lasers (FELs). We present results of emittance measurements carried out at SwissFEL, a compact x-ray FEL facility at the Paul Scherrer Institute in Switzerland, including a description of the novel high-resolution measurement techniques and the optimization procedure. We obtained slice emittance values at the undulator entrance down to 200 nm for an electron beam with a charge of 200 pC and an rms duration of 30–40 fs. Furthermore, we achieved slice emittances as low as 100 nm for 10 pC beams with few fs duration. These values set new standards for electron linear accelerators. The quality, verification, and control of our electron beams allowed us to generate high-power FEL radiation for a wavelength as short as 0.1 nm using an electron beam with an energy of only 6 GeV. The emittance values demonstrated at SwissFEL would allow producing hard x-ray FEL pulses with even lower-energy beams, thus paving the way for even more compact and cost-effective FEL facilities.