Superconductivity is one of the most important research fields in condensed matter physics. The rapid development of material preparation technology in last few years has made the experimental study of low-dimensional physical superconducting properties feasible. This article gives a brief introduction on superconductivity and technology of low-dimensional material fabrication, and mainly focuses on the experimental progress in electrical transport studies on one-and two-dimensional superconductors, especially the results from our group. As for one-dimensional superconductivity, we review the superconductivities in single crystal Bi nanowires, crystalline Pb nano-belts, and amorphous W nanobelts, and the proximity effects in superconducting nanowires, metallic nanowires, and ferromagnetic nanowires. Surface superconductivity is revealed for crystalline Bi nanowire. The step-like voltage platforms in V-I curves are observed in Pb nano-belts and may be attributed to phase slip centers. Besides, vortex glass (VG) phase transition is discovered in amorphous W nano-belts. Inverse proximity effect is detected in crystalline Pb nanowires with normal electrodes, and proximity induced mini-gap is found in crystalline Au nanowire with superconducting electrodes. Furthermore, in crystalline ferromagnetic Co nanowire contacted by superconducting electrodes, unconventional long range proximity effect is observed. As for two-dimensional superconductivity, we review the superconductivities in Pb thin films on Si substrates, 2 atomic layer Ga films on GaN substrates, and one-unit-cell thick FeSe film on STO substrates grown by molecular beam epitaxy (MBE) method. By both in situ scanning tunneling microscopy/spectroscopy and ex situ transport and magnetization measurements, the two-atomic-layer Ga film with graphene-like structure on wide band-gap semiconductor GaN is found to be superconducting with Tc up to 5.4 K. By direct transport and magnetic measurements, the strong evidences for high temperature superconductivities in the 1-UC FeSe films on insulating STO substrates with the onset Tc and critical current density much higher than those for bulk FeSe are revealed. Finally, we give a summary and present a perspective on the future of low dimensional superconductors.