The regular logistic map was introduced in 1960s, served as an example of a complex system, and was used as an instrument to demonstrate and investigate the period-doubling cascade of bifurcations scenario of transition to chaos. The first fractional generalization was introduced in 2002. The two generalizations which are based on the continuous and discrete Caputo fractional calculus are the fractional logistic map (FLM) and the fractional difference logistic map (FDLM). They are well investigated. The finite time evolution of the FLM and FDLM is characterized by cascade of bifurcations-type trajectories (CBTTs) and strong dependance on the initial conditions and the number of iterations. The map's asymptotic behavior is described by the conditions of the asymptotic stability, the rate of convergence to the asymptotically periodic points, asymptotic bifurcation points, and the transition to chaos. The maps were used to show numerically that the fractional Feigenbaum constant δ exists and is equal to its regular value. Fractional generalizations may also be used to naturally introduce the 2D and 3D logistic maps. Applications of the FLM and the FDLM include cryptography, distribution of ageing, population biology, etc. One of the most important remaining problems is a theoretical analysis of the asymptotic universality in fractional dynamics which could be based on the derived equations defining the asymptotically periodic and bifurcation points.