A key requirement for many practical applications of the new class of high Tc superconductors is the ability to sustain high current densities in the presence of magnetic fields. The new materials present two substantially different behaviors. For Ba2YCu3O7 critical current behavior of polycrystalline material is dominated by intergranular transport. We will review a model for transport involving the effects of the correlation of defects with grain boundary orientation, the presence of impurities at the grain boundaries, measurements of tunneling characteristics through boundaries, and the effect of grain alignment on critical current. The case is substantially different in the 84 K superconducting phase Bi2.2Sr2Ca0.8Cu2O8+δ where there is a very strong dependence of critical current densities at 30-84 K on magnetic field for single crystals. There is a lack of threshold behavior in the I-V characteristics in finite applied fields larger than Hc1(T) which is attributable to flux motion. This suggests that in the Bi material there is insufficient pinning of vortices in the intrinsic material to prevent them from moving in the presence of an applied transport current, causing energy dissipation. These results are in agreement with observations of flux lattice melting in single crystals of Ba2YCu3O7 and Bi2.2Sr2Ca0.8Cu2O8+δ. The implications of these various factors on potential applications of these materials will be discussed.