<p indent=0mm>Firstly proposed in 1946 by George Gamow, the hot big bang theory is now the most widely accepted cosmological model of the Universe, where the Universe expanded from a very high-density state dominated by radiation. The theory has been vindicated by the observation of the cosmic microwave background, our emerging knowledge of the large-scale structure of the Universe, and the rough consistency between calculations and observations of primordial abundances of the lightest elements in nature: Hydrogen, helium, and lithium. Primordial big-bang nucleosynthesis (BBN) began when the Universe was <sc>3 minutes</sc> old and ended less than half an hour later when nuclear reactions were quenched by the low temperature (about <sc>10<sup>9</sup> K,</sc> i.e., ~1 GK, or <italic>kT</italic><sc>~100 keV)</sc> and density conditions in the expanding Universe. Only the lightest nuclides (<sup>2</sup>H, <sup>3</sup>He, <sup>4</sup>He, and <sup>7</sup>Li) were synthesized in appreciable quantities through BBN, and these relics provide us a unique window on the early Universe. Constrained by the baryon density determined by the Wilkinson Microwave Anisotropy Probe (WMAP) and PLANCK satellites, the primordial abundances of <sup>2</sup>H (referred to as D hereafter) and <sup>4</sup>He inferred from observational data are in good general agreement with predictions; however, the standard BBN theory overestimates the primordial <sup>7</sup>Li abundance by about a factor of three to four. This significant deviation is the so-called “cosmological lithium problem.” Over about two decades, many attempts to resolve this discrepancy using conventional nuclear physics have been unsuccessful yet. The possible solutions to this unresolved problem have been suggested, which can be mainly categorized into three groups, according to which part of the preceding analysis is called into question: (1) Astrophysical solutions revise the measured primordial lithium abundance; (2) solutions beyond the standard model invoke new particle physics or nonstandard cosmological physics; (3) nuclear physics solutions alter the reaction flow into and out of mass-7. This paper systematically reviews three major aspects of this long-pending cosmological lithium problem. It includes the scientific background, motivation, possible solutions and attempts made in the past. Here, we mainly focus on the nuclear physics solutions, which might solve this problem in an elegant way. In this review, the goals of different research directions are summarized and evaluated, and the promising direction is guided for future investigations.
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