We examine to what extent nonlinear input-disturbance systems that are connected by a simulation relation share certain controllability properties. We derive several results that fit within the paradigm that if there is a simulation relation of a system A by a system B, and if system A has a specified controllability property, then system B has that same property. As expected, one can only turn the paradigm into actual theorems by imposing appropriate assumptions on the systems and/or the simulation relation. We prove three such results. The first result we obtain deals with the property of complete controllability, where we impose minimal assumptions on the input-disturbance systems but require that the simulation relation be the graph of a smooth surjection between the systems' state spaces that satisfies a certain compactness condition. The second result deals with a somewhat weaker notion of controllability modulo the kernel of a linear mapping, where it is assumed that system B is "almost linear," but where the simulation relation is the zero set of a smooth mapping of a specific form (but is not necessarily the graph of a function). The third result (and the most difficult to prove) also deals with the property of complete controllability and, while imposing minimal assumptions on the input-disturbance systems, allows the simulation relation to be the zero set of a smooth function (and so not necessarily a graph), though other somewhat restrictive assumptions do have to be imposed. We conclude with several examples to illustrate our results.