This dissertation focuses on the non-equilibrium dynamics of soft matter systems. In the first part, I explore the time dependent microstructural evolution of aging aqueous colloidal suspensions of Laponite clay using falling ball viscometry and optical tweezer-based active microrheology. Freely falling objects (steel balls, diameters 2 mm – 6 mm) in a falling ball viscometer containing an aging clay suspension are tracked to probe the fragile microscopic suspension structures. A simple model is proposed to explain the unique motion of the falling balls while accounting for the non-Newtonian rheology of the aging suspension. To further investigate the particle-scale dynamics in aging Laponite suspensions, the motion of a micron sized bead trapped optically inside the suspension is studied by exploiting an active micro-rheology technique. In this experiment, an optical tweezer set-up traps a polystyrene bead while the aging suspension in which it is trapped is subjected to oscillatory displacements. The resulting force response on the trapped bead is measured to examine the underlying microscopic structure of the Laponite suspension. We also use optical tweezer based passive microrheology to study the dynamics of aqueous suspensions of the heterogeneously charged Laponite particles while applying AC electric fields.  Combining data acquired using a rheometer and an optical tweezer, we report a non-monotonic variation of the suspension aging time with increasing AC field amplitudes. In the second part of my talk, I shall present our studies to investigate the development of dynamical heterogeneities (DHs) in dense colloidal and granular systems. Our measurements are performed on two systems: colloidal suspensions of micron sized poly(N-isopropylacrylamide) or PNIPAM particles and a granular system of Teflon balls (diameter 3 mm – 7 mm). We synthesize batches of PNIPAM particles with varying size polydispersity indices (PDIs). Near the jamming transition, the effect of particle size PDI on the dynamics of PNIPAM particles is studied. In order to comprehend the observed dynamics, I conduct molecular dynamics simulations of dense suspensions. In both experiments and simulations, we show that DHs disappear as PDI increases. The dynamics of granular balls is investigated using an electromagnetic shaker while changing the discrete size distribution of the balls. Growth of DHs is observed with increasing area fraction as we approach the random close packing fraction for balls of two different size distributions.