Answering an open question affirmatively it is shown that every ergodic invariantmeasure of a mean equicontinuous (i.e.mean-L-stable) system has discretespectrum.Dichotomy results related to mean equicontinuity and meansensitivity are obtained when a dynamical system is transitive or minimal.Localizing the notion of mean equicontinuity,notions of almost mean equicontinuityand almost Banach mean equicontinuity are introduced.It turns outthat a system with the former property may have positive entropy and meanwhilea system with the later property must have zero entropy.(this is a jointwork with Li and Tu).