FracSym: automated symbolic computation of Lie symmetries of fractional differential equations. In this paper, we present an algorithm for the systematic calculation of Lie point symmetries for fractional order differential equations (FDEs) using the method as described by E. Buckwar and Y. Luchko [J. Math. Anal. Appl. 227, No. 1, 81–97, Art. No. AY986078 (1998; Zbl 0932.58038)] and R. K. Gazizov et al. [“Continuous transformation groups of fractional differential equations”, Vestn. USATU 9, No. 21, 125–135 (2007); “Symmetry properties of fractional diffusion equations”, Phys. Scr. 2009, T136, Article ID 014016, 6 p. (2009; doi:10.1088/0031-8949/2009/T136/014016); in: Nonlinear science and complexity. Based on the 2nd conference on nonlinear science and complexity, NSC ’08, Porto, Portugal, 2008. Berlin: Springer. 51–59 (2011; Zbl 1217.37066)]. The method has been generalised here to allow for the determination of symmetries for FDEs with n independent variables and for systems of partial FDEs. The algorithm has been implemented in the new MAPLE package FracSym [the authors, Comput. Phys. Commun. 184, No. 3, 1045–1063 (2013; Zbl 1306.65267)] which uses routines from the MAPLE symmetry packages DESOLVII [K. T. Vu et al., Comput. Phys. Commun. 183, No. 4, 1044–1054 (2012; Zbl 1308.35002)] and ASP [the authors, Comput. Phys. Commun. 184, No. 3, 1045–1063 (2013; Zbl 1306.65267)]. We introduce FracSym by investigating the symmetries of a number of FDEs; specific forms of any arbitrary functions, which may extend the symmetry algebras, are also determined. For each of the FDEs discussed, selected invariant solutions are then presented.