This paper provides a comprehensive review of fractional Black-Scholes equations (FBSEs) and their solution techniques, focusing on European option pricing. The classical Black-Scholes (B-S) equation, which models the dynamics of option prices, has limitations due to its assumption of constant volatility and Gaussian distribution. To address these limitations, FBSEs incorporating fractional derivatives (FDs) have been developed to better capture the non-local and long-range dependence characteristics observed in financial markets. The paper discusses various FBSEs, including those based on modified Lévy processes (such as KoBoL, CGMY, and FMLS processes) and time-space fractional B-S equations. It also explores different solution techniques, such as analytic solutions using Fourier transforms and numerical methods like finite difference schemes, spectral methods, and wavelet transforms. The paper highlights the advantages of FBSEs in accurately modeling extreme events and complex market phenomena, providing a robust basis for derivative pricing and risk management. The review aims to enrich the research methods in finance and facilitate interdisciplinary collaboration.This paper provides a comprehensive review of fractional Black-Scholes equations (FBSEs) and their solution techniques, focusing on European option pricing. The classical Black-Scholes (B-S) equation, which models the dynamics of option prices, has limitations due to its assumption of constant volatility and Gaussian distribution. To address these limitations, FBSEs incorporating fractional derivatives (FDs) have been developed to better capture the non-local and long-range dependence characteristics observed in financial markets. The paper discusses various FBSEs, including those based on modified Lévy processes (such as KoBoL, CGMY, and FMLS processes) and time-space fractional B-S equations. It also explores different solution techniques, such as analytic solutions using Fourier transforms and numerical methods like finite difference schemes, spectral methods, and wavelet transforms. The paper highlights the advantages of FBSEs in accurately modeling extreme events and complex market phenomena, providing a robust basis for derivative pricing and risk management. The review aims to enrich the research methods in finance and facilitate interdisciplinary collaboration.