Glass transitions are commonly observed in many systems including polymers. The glassy states, which could be obtained by cooling liquid states of polymers at high temperature down to lower temperature, show characteristic common properties such as memory and rejuvenation effects. In this paper, we show our recent experimental results including glass transition dynamics of thin polymer films and aging phenomena in polymer glasses after giving an overview of glass transition and glassy dynamics. For thin films of polystyrene, dielectric measurements revealed that the glass transition temperature decreases with decreasing film thickness and the α-relaxation process becomes faster with decreasing film thickness. For poly(methyl methacrylate), aging phenomena have been investigated using dielectric measurements. The memory and rejuvenation effects could be observed for ac-dielectric susceptibility in the temperature cycling mode and are found to be almost the same to those observed in ac-magnetic susceptibility of spin glasses.