This thesis describes low-temperature transport measurements in low-dimensional systems fabricated in high-mobility GaAs/AlGaAs heterostructures. These low-dimensional systems are formed by electrostatically constricting the electrons in the two-dimensional electron gas at the interface of the heterostructure, by applying a voltage to a pair of metallic gates known as a split-gate. At low temperatures the electrical conduction occurs without scattering. The aim is to measure the thermal conductance of these one-dimensional ballistic conductors. The thermal conductance of a split-gate device was measured as a function of gate voltage, over a wide range of temperatures and in the absence of a magnetic field. The electrons on one side of the constriction were heated with an electric current, and the temperature drop across the split-gate was measured using the thermopower of another split-gate. The measurements show that the thermal conductance displays plateaux corresponding to the one-dimensional subbands, confirming previous results. The design of the samples allows a quantitative test of the Wiedemann-Franz law in one-dimensional constrictions. The results strongly suggest that the Wiedemann-Franz law is satisfied, and new information was obtained regarding the anomaly in the conductance known as 0.7 structure, which can provide a new insight into the nature of this anomaly. It is found that the thermal conductance corresponding to the anomaly is suppressed with respect to the value expected from a single-particle picture.