Abstract: Two-dimensional (2D) honeycomb ferromagnets, such as monolayer chromium trihalides, are predicted to behave as topological magnon insulators, characterized by an insulating bulk and topologically protected edge states, giving rise to a thermal magnon Hall effect. Here we report the behavior of the topological magnons in monolayer CrI3 under external stimuli, including biaxial and uniaxial strain, electric gating, as well as in-plane and out-of-plane magnetic field, revealing that one can thereby tailor the magnetic states as well as the size and the topology of the magnonic bandgap. These findings broaden the perspective of using 2D magnetic materials to design topological magnonic devices.