A continuum $X$ is called a \emph{common model} for a class $\mathcal{C}$ of continua if every member of $\mathcal{C}$ is a continuous image of $X$. One of the natural questions in continuum theory is whether a given class of continua admits a common model, and if not, how the non-existence of common models can be established.

In this talk, we will discuss several recent results concerning the non-existence of common models for classes of continua arising in hyperspace theory and the theory of indecomposable continua. The main tool is a recent theorem on meandering continua, which provides a general method for establishing non-existence results.

As applications, we will present new classes of continua associated with Whitney properties and Whitney reversible properties, together with several classes related to indecomposable continua, and show that these classes do not admit common models.