q Fractional Calculus and Equations This nine chapter monograph introduces a rigorous investigation of q difference operators in standard and fractional settings. It starts with elementary calculus of q differences and integration of Jackson’s type before turning to q difference equations. The existence and uniqueness theorems are derived using successive approximations leading to systems of equations with retarded arguments. Regular q Sturm Liouville theory is also introduced; Green’s function is constructed and the eigenfunction expansion theorem is given. The monograph also discusses some integral equations of Volterra and Abel type as introductory material for the study of fractional q calculi. Hence fractional q calculi of the types Riemann Liouville; Grünwald Letnikov; Caputo; Erdélyi Kober and Weyl are defined analytically. Fractional q Leibniz rules with applications in q series are also obtained with rigorous proofs of the formal results of Al Salam Verma which remained unproved for decades. In working towards the investigation of q fractional difference equations; families of q Mittag Leffler functions are defined and their properties are investigated especially the q Mellin Barnes integral and Hankel contour integral representation of the q Mittag Leffler functions under consideration the distribution asymptotic and reality of their zeros establishing q counterparts of Wiman’s results. Fractional q difference equations are studied; existence and uniqueness theorems are given and classes of Cauchy type problems are completely solved in terms of families of q Mittag Leffler functions. Among many q analogs of classical results and concepts q Laplace q Mellin and q2 Fourier transforms are studied and their applications are investigated 1 Preliminaries 2 q Difference Equations 3 q Sturm Liouville Problems 4 Riemann Liouville q Fractional Calculi 5 Other q Fractional Calculi 6 Fractional q Leibniz Rule and Applications 7 q Mittag Leffler Functions 8 Fractional q Difference Equations 9 Applications of q Integral Transforms