This review presents a comprehensive analysis of the Fractional Black-Scholes (FBSE) equations and their solution techniques for pricing European options. The classical Black-Scholes (B-S) equation, developed in the 1970s, is based on several assumptions, including a frictionless market and constant volatility, which limit its applicability in real-world financial markets. The introduction of fractional calculus has enabled the development of FBSEs, which incorporate non-local properties and fractal characteristics observed in financial markets. These equations offer a more flexible representation of market behavior by accounting for long-range dependence, heavy-tailed distributions, and multifractality, leading to better modeling of extreme events and complex market phenomena. FBSEs are more accurate in depicting price fluctuations in actual financial markets, providing a more reliable basis for derivative pricing and risk management. This paper reviews various FBSEs, including space, space-time, and time fractional B-S equations, and discusses their associated solution techniques. These techniques include analytical methods such as the Laplace transform and integral transforms, as well as numerical methods like finite difference schemes, spectral methods, and wavelet-based approaches. The review also highlights the importance of understanding the underlying assumptions and properties of these equations, as well as the practical implications for financial modeling and risk management. The paper emphasizes the need for a systematic and comprehensive overview of FBSEs and their solution techniques, as they play a crucial role in characterizing heavy-tailed phenomena in option pricing. The review also discusses the challenges in deriving analytical solutions and the development of efficient numerical methods for solving these equations. Overall, the paper aims to provide a valuable resource for researchers in finance, enabling them to gain a deeper understanding of the practical applications of these equations in financial markets.This review presents a comprehensive analysis of the Fractional Black-Scholes (FBSE) equations and their solution techniques for pricing European options. The classical Black-Scholes (B-S) equation, developed in the 1970s, is based on several assumptions, including a frictionless market and constant volatility, which limit its applicability in real-world financial markets. The introduction of fractional calculus has enabled the development of FBSEs, which incorporate non-local properties and fractal characteristics observed in financial markets. These equations offer a more flexible representation of market behavior by accounting for long-range dependence, heavy-tailed distributions, and multifractality, leading to better modeling of extreme events and complex market phenomena. FBSEs are more accurate in depicting price fluctuations in actual financial markets, providing a more reliable basis for derivative pricing and risk management. This paper reviews various FBSEs, including space, space-time, and time fractional B-S equations, and discusses their associated solution techniques. These techniques include analytical methods such as the Laplace transform and integral transforms, as well as numerical methods like finite difference schemes, spectral methods, and wavelet-based approaches. The review also highlights the importance of understanding the underlying assumptions and properties of these equations, as well as the practical implications for financial modeling and risk management. The paper emphasizes the need for a systematic and comprehensive overview of FBSEs and their solution techniques, as they play a crucial role in characterizing heavy-tailed phenomena in option pricing. The review also discusses the challenges in deriving analytical solutions and the development of efficient numerical methods for solving these equations. Overall, the paper aims to provide a valuable resource for researchers in finance, enabling them to gain a deeper understanding of the practical applications of these equations in financial markets.