I will present a construction of universal mixers in all dimensions, i.e., incompressible flows that mix to arbitrarily small scales general solutions to the corresponding transport equation.  While no universal mixer can have a uniform mixing rate for all measurable initial data, these flows are also almost-universal exponential mixers in the sense that they achieve exponential-in-time mixing (which is the optimal rate) for all initial data with at least some degree of regularity.  The constructed flows are time-dependent with an alternating cellular structure, and exist on tori as well as on bounded domains in Euclidean spaces.  I will also present numerical evidence of exponential mixing by a different class of flows, alternating shear flows on two-dimensional tori.