In this thesis, we will show results on two different self-interacting random walk models on $\ints$.

First, we observe the frog model, an infinite system of interacting random walks, on $\ints$ with an asymmetric underlying random walk. For certain initial frog distributions we construct an explicit formula for the moments of the leftmost visited site, as well as their asymptotic scaling limits as the drift of the underlying random walk vanishes. We also provide conditions in which the lower bound can be scaled to converge in probability to the degenerate distribution at 1 as the drift vanishes.

Then, we state and prove a theorem on the bound of the number of favorite (i.e., most visited) sites for the symmetric persistent random walk on $\ints$, a discrete-time process typified by the correlation of its directional history. This is a generalization of a result by T\'{o}th used to partially prove a longstanding conjecture by Erd\H{o}s and R\'{e}v\'{e}sz.

\par

We conclude with examples of potential future directions of research in these problems and related topics.