In modern semiconductor devices, only the charge of electrons is being utilized for the manipulation and transport of information. The spin degree of freedom of the electrons has not been exploited in any commercial semiconductor application so far. People in the field of Spintronics are trying to find new or improved functionalities and applications that are based on the spin of electrons, instead of - or in addition to - its charge. Some of these new ideas are based on the binary nature of the electron spin, which makes that it could serve as an elementary digital bit. When the spin state can be transported and manipulated in space and time, the spin could be used for new calculation schemes, for example. In addition, it has been predicted that the motion of electrons through materials can be affected by acting on their spin state. Hence, the spin could potentially function as a "knob" for tuning the conductivity of electrons. From the above it follows that it is important to understand mechanisms that allow to manipulate the spin dynamics (i.e. the spin state of electrons). The best-known mechanism to affect the spin dynamics is by an external magnetic field, i.e., via Zeeman coupling to the spin. Another potentially important mechanism is Rashba spin-orbit interaction (SOI). The interesting and distinguishing feature of Rashba SOI is, that it can be tuned by electrostatic means. This might make it easier to control the spin dynamics on very small length scales, in comparison to Zeeman coupling. Rashba SOI is also known, however, to be the main cause of spin relaxation in many 2D electron systems, thereby destroying any information that is encrypted on the spin. For the field of Spintronics it is therefore important to understand the effects of Rashba SOI, and the interplay between Rashba SOI and Zeeman coupling, on electron transport and on the spin dynamics. In this Thesis we have studied partly theoretically and partly experimentally the influence of Rashba SOI on low temperature electron transport in mesoscopic (phase-coherent) systems, such as Hall bars and ring structures. In addition, we have experimentally investigated how the competition between Rashba SOI and Zeeman coupling affects the spin dynamics in 2D electron gases, and if/how information about the resulting spin dynamics, such as the spin-relaxation time, can be obtained from "simple" transport measurements. Finally, we have studied the effect of the interplay between Rashba SOI and Zeeman coupling on the breaking of time reversal symmetry in 2D electron gases.