  The first half of this paper explores the origin of systematic biases in the
measurement of weak gravitational lensing. Compared to previous work, we expand
the investigation of PSF instability and fold in for the first time the effects
of non-idealities in electronic imaging detectors and imperfect galaxy shape
measurement algorithms. Together, these now explain the additive A(l) and
multiplicative M(l) systematics typically reported in current lensing
measurements. We find that overall performance is driven by a product of a
telescope/camera's *absolute performance*, and our *knowledge about its
performance*.
  The second half of this paper propagates any residual shear measurement
biases through to their effect on cosmological parameter constraints. Fully
exploiting the statistical power of Stage IV weak lensing surveys will require
additive biases A<1.8e-12 and multiplicative biases M<4.0e-3. These can be
allocated between individual budgets in hardware, calibration data and
software, using results from the first half of the paper.
  If instrumentation is stable and well-calibrated, we find extant shear
measurement software from GREAT10 already meet requirements on galaxies
detected at S/N=40. Averaging over a population of galaxies with a realistic
distribution of sizes, it also meets requirements for a 2D cosmic shear
analysis from space. If used on fainter galaxies or for 3D cosmic shear
tomography, existing algorithms would need calibration on simulations to avoid
introducing bias at a level similar to the statistical error. Requirements on
hardware and calibration data are discussed in more detail in a companion
paper. Our analysis is intentionally general, but is specifically being used to
drive the hardware and ground segment performance budget for the design of the
European Space Agency's recently-selected Euclid mission.
