The work described in this thesis consists of an investigation into the behaviour of the critical current density (Jc) of several different high temperature superconductors (HTSCs) as a function of temperature and applied magnetic field. The focus of this research has been to investigate the discrepancies which generally exist between magnetic and transport measurements on HTSCs. In order to do this a number of systems were selected with different weak link strengths, overall alignment and pinning characteristics. Systematic studies were carried out on these systems using both transport and magnetic techniques. The results obtained were compared to obtain a coherent picture of the relation between critical currents in HTSCs, their granularity and structure, and how this accounts for the differences in magnetic and transport measurements. The Jc of granular bulk sintered YBCO was measured as a function of magnetic field and orientation, and attempts made to fit the results obtained to theory. The results obtained indicate that hysteresis of Jc with field cannot be explained by flux trapping alone. Jc measurements were carried out on the 'hub-and-spoke' (H-S) grains of melt-processed YBCO thick films. These indicated that within each H-S grain the current is constrained to radial paths through the centre of the grain. This has major implications for the analysis of any measurements carried out on these samples, as well as for applications, as only a small fraction of the sample carries the applied current. Silver-clad tapes of Tl:1223 and Tl:2223 were measured in different temperatures, applied magnetic fields and orientations. The variation of Jc along the tape length was also measured to provide an estimate of their homogeneity. It was found that the processing route used in the production of these tapes had not aligned the superconductor within them, and also that the tape properties were not homogenous along their length. This probably arises from the crystal structure of the thallium materials. The Jcs of melt-processed thick films of Bi:2212 on silver substrates were measured as a function of temperature and applied magnetic field, and compared with results obtained from magnetic measurements obtained from a VSM. Attempts were made to fit these results to theory. The results obtained show clear scaling behaviour of Jc with T at constant B and imply that the variation in the form of the Jc versus T curves is due simply to the suppression of Tc with increasing applied field.