Tensor models exhibit a melonic large-N limit. They were first introduced in zero dimension in the context of random geometry and quantum gravity. They were then generalized in d dimensions where they give rise in the infrared to a new kind of CFTs, analytically accessible, called melonic CFTs. In the first part of the talk, I will briefly review melonic CFTs and present in particular the long-range bosonic O(N)^3 model. In the second part of the talk, I will show that this model respects the F-theorem at large N. The F-theorem states that in three dimensions the sphere free energy of a CFT must decrease between ultraviolet and infrared fixed points. It has been proven for unitary CFTs but only hints of unitarity were found at large N for the long-range O(N)^3 model and it is non-unitary at finite N. This model thus provides a new non-trivial example of the F-theorem.