**Summary:** This paper explores the scaling limits of loop-erased random walks (LERW) and uniform spanning trees (UST) in two dimensions. The authors investigate the behavior of these processes on increasingly fine grids as the mesh size approaches zero. While the existence of scaling limits remains unproven, they establish that subsequential limits exist and have specific properties. For LERW, the scaling limit is shown to be a simple path, and for UST, the scaling limit is a topological tree that is dense in the plane. The paper conjectures that these processes are conformally invariant, a property that, if true, would imply an explicit construction of their scaling limits. The authors use a Löwner differential equation with Brownian motion as the parameter to describe the LERW scaling limit. They also show that this process is related to the boundary of macroscopic critical percolation clusters. For UST, the scaling limit is shown to be a tree that is dense in the plane, and the paper proves that the trunk of the UST scaling limit is a topological tree. The authors also discuss the relationship between UST and domino tilings, and how properties of UST can be derived using Kenyon's machinery. The paper concludes with a discussion of Schramm-Loewner Evolution (SLE) and its connection to the scaling limits of LERW and UST. It shows that SLE with specific parameters can describe the scaling limits of these processes, and that these processes are related to critical percolation and the Peano curve. The authors also discuss the implications of conformal invariance for these processes and the potential for future proofs of the conformal invariance conjecture.**Summary:** This paper explores the scaling limits of loop-erased random walks (LERW) and uniform spanning trees (UST) in two dimensions. The authors investigate the behavior of these processes on increasingly fine grids as the mesh size approaches zero. While the existence of scaling limits remains unproven, they establish that subsequential limits exist and have specific properties. For LERW, the scaling limit is shown to be a simple path, and for UST, the scaling limit is a topological tree that is dense in the plane. The paper conjectures that these processes are conformally invariant, a property that, if true, would imply an explicit construction of their scaling limits. The authors use a Löwner differential equation with Brownian motion as the parameter to describe the LERW scaling limit. They also show that this process is related to the boundary of macroscopic critical percolation clusters. For UST, the scaling limit is shown to be a tree that is dense in the plane, and the paper proves that the trunk of the UST scaling limit is a topological tree. The authors also discuss the relationship between UST and domino tilings, and how properties of UST can be derived using Kenyon's machinery. The paper concludes with a discussion of Schramm-Loewner Evolution (SLE) and its connection to the scaling limits of LERW and UST. It shows that SLE with specific parameters can describe the scaling limits of these processes, and that these processes are related to critical percolation and the Peano curve. The authors also discuss the implications of conformal invariance for these processes and the potential for future proofs of the conformal invariance conjecture.