The deformation behavior and structure changes of magnesium alloy AZ31 were studied in compression at temperatures ranging from 523K to 673K and at a strain rate of 3 x 10-3 s-1. They depend sensitively on deformation temperature. At high temperatures, grain fragmentation takes place due to frequent formation of kink bands initially at corrugated grain boundaries and then in grain interiors, followed by full development of new grains in high strain. At lower temperatures, in contrast, twinning takes place in rather coarse grains and kink bands are formed mainly in finer original ones in low strain. It is concluded that new grain evolution can be controlled by a deformation-induced continuous reaction resulting in grain fragmentation by kink bands, i.e. continuous dynamic recrystallization (cDRX). The latter is discussed comparing with conventional, i.e. discontinuous, DRX.