  Understanding the topological aspects of the band structure of solids has
fundamentally changed our appreciation of their properties. The layered, van
der Waals transition-metal pentatelluride ZrTe$_5$ has proven on numerous
occasions to be an excellent candidate for the study of controllable
topological phase transitions. Here, we investigate the topological phase
diagrams of monolayer and bilayer forms of ZrTe$_5$, under mechanical
deformations using \textit{ab initio} techniques. We find that mechanical
deformation can close the monolayer's topological gap, while the bilayer
exhibits richer phase diagram, including both topological insulating, trivial
metallic and insulating phases. The bilayer is predicted to be on the
topological phase boundary. We also address the preparation of monolayers,
using \emph{ab initio} simulations and experimental scanning tunneling
microscopy measurements. We confirm that while monolayer ZrTe$_5$ is difficult
to exfoliate without compromising its crystalline structure, bilayers offer a
more stable alternative, revealing the complexities and limitations of using
gold substrates for monolayer exfoliation.
