We study periodically driven pure Kitaev model and ferromagnetic phase of the Kitaev-Heisenberg model on the honeycomb lattice by off-resonant linearly and circularly-polarized lights at zero magnetic field. Using a combination of linear spin wave and Floquet theories, we show that the effective time-independent Hamiltonians in the off-resonant regime map onto the corresponding anisotropic static spin model, plus a tunable photoinduced magnetic field along the [111] direction, which precipitates Floquet topological magnons and chiral magnon edge modes. They are tunable by the light amplitude and polarization. Similarly, we show that the thermal Hall effect induced by the Berry curvature of the Floquet topological magnons can also be tuned by the laser field. Our results pave the way for ultrafast manipulation of topological magnons in irradiated Kitaev magnets, and could play a pivotal role in the investigation of ultrafast magnon spin current generation in Kitaev materials.Introduction.-Topological band theory of solid-state materials has dominated many aspects of condensedmatter physics over the past decade [1, 2]. The original concept of topological band theory is rooted in insulating electronic systems possessing a nontrivial gap in their energy band structures. They are characterized by the appearance of gapless chiral edge electron modes traversing the bulk gap, which are topologically protected by the Chern number or the Z 2 index of the bulk bands [1, 2].Generally, the concept of topological band structure is independent of the statistical nature of the quasiparticle excitations and therefore is not restricted to insulating electronic systems. Recently, there has been a tremendous interest in the topological properties of spin excitations in insulating quantum magnets. In fact, bosonic topological spin excitations (magnons and triplons) have been studied in many different insulating quantum magnets [3][4][5][6][7][8][9][10][11][12][13][14], and the appearance of chiral edge modes and bulk Chern number have been demonstrated [4][5][6]. Recently, bosonic topological spin excitations mimicking electronic topological insulators have been experimentally observed in kagome ferromagnet Cu(1,3-bdc) [9], dimerized quantum magnet SrCu 2 (BO 3 ) 2 [10], and honeycomb ferromagnet CrI 3 [11].The Mott-insulating honeycomb Kitaev magnets are currently of great interest [15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33]. Candidate Kitaev materials include Na 2 IrO 3 and α-RuCl 3 [26][27][28][29][30]. Recently, topologically protected spin waves have been predicted in the fully-polarized phase of the pure Kitaev model [12] and the Kitaev-Heisenberg model [13] at high magnetic field. In the former, the topological magnons and chiral edge states present in linear spin-wave approximation survive magnon-magnon interactions and therefore are robust [12]. Indeed, the manipulation of topological magnons and magnon spin currents is essential for their practical applications in ultrafast magnetic data s...