This paper addresses the problems of observability analysis and state observer design for nonlinear systems based on the study of linear systems which mimic their dynamics. It is shown that, if a nonlinear system can be related to a linear system through an appropriate transformation of the state variables, the observability analysis of the linear system can determine observability or non-observability of the nonlinear system, depending on the properties of the state transformation function. Conditions are also derived for state observers for the linear systems to double as state observers for the original nonlinear systems, as well as retaining exponential stability properties. To illustrate the usefulness of those results, several application examples are detailed.