In this review paper, we begin by introducing the fundamental concepts of superconductivity, laying the groundwork for understanding its principles and applications. We then delve into the scientific advantages of one-dimensional (1D) superconductors over three-dimensional (3D) superconductors, highlighting the main significant enhancement in the upper critical field, which can increase by two orders of magnitude. This feature is crucial for advancing the technological performance of superconducting high-field magnets. In spite of the technological advantages of 1D superconductivity, we discuss the manufacturing challenges associated with 1D superconducting nanowires, including issues related to synthesis, scalability, and material quality, etc. Overcoming these hurdles is essential for realizing the potential of 1D superconductivity in practical applications. Then we review research progress in 1D superconductivity, with a particular focus on the search for Majorana zero modes. This involves examining the intriguing possibilities of superconductivity in topological systems, where the existence of Majorana modes could lead to breakthroughs in quantum computing and other technologies. Through this review, we aim to provide a short overview of the progress and challenges in this exciting field of study, transforming from conventional 1D superconductivity to topological superconductivity.