The intention of this thesis is to explore non-compact objects evolving under geometric flows without any prescribed boundary control. In particular, we focus on Ricci flow and curve shortening flow defined on open manifolds. We consider the uniqueness of the Cauchy problem for smooth properly embedded curves evolving under curve shortening flow. We construct an example to show that the class of smooth properly embedded solutions is too large to expect uniqueness, and in turn introduce a new suitable subclass for which we conjecture uniqueness when our ambient space is flat. We then show that even within this subclass of solutions, we can have non-uniqueness of the Cauchy problem in general ambient surfaces. Finally, we give a partial characterisation of those ambient surfaces which exhibit this non-uniqueness. To complement these results, we also consider low dimensional Ricci flow spacetimes. We prove that, for a complete (2 + 1)-dimensional Ricci flow spacetime, the spatial-slice at any later time must contain (under the flow of the time vector field) the spatial-slice at any earlier time. We then prove, after imposing a necessary regularity condition on such a complete Ricci flow spacetime, that all of its spatial-slices must agree with one another, and our Ricci flow spacetime is isomorphic to a classical Ricci flow on a fixed ambient surface.