[en] Many future linear collider designs require beams with very small transverse emittances and large emittance ratios ε_x much-gt ε_y. In this paper, we will discuss issues associated with the preservation of these small emittances during the acceleration of the beams. The primary sources of transverse emittance dilution in a high energy linear accelerator are the transverse wakefields, the dispersive errors, RF deflections, and betatron coupling. We will discuss the estimation of these effects and the calculation of tolerances that will limit the emittance dilution with a high degree of confidence. Since the six-dimensional emittance is conserved and only the projected emittances are increased, these dilutions can be corrected if the beam has not filamented (phase mixed). We discuss methods of correcting the dilutions and easing the tolerances with beam-based alignment and steering techniques, and non-local trajectory bumps. Finally, we discuss another important source of luminosity degradation, namely, pulse-to-pulse jitter