The thesis concerns questions related to the dynamics of non-locally compact Polish groups. Chapter 2 defines a projective Fraïssé family whose limit approximates the universal Knaster continuum. The universal Knaster continuum is an indecomposable compact, connected, metrizable space. The projective Fraisse family is such that the group $\aut(\mathbb{K})$ of automorphisms of the Fraïssé limit is a dense subgroup of the group, $\Homeo(K)$, of homeomorphisms of the universal Knaster continuum. It is shown that both $\aut(\mathbb{K})$ and $\Homeo(K)$ have universal minimal flow homeomorphic to the universal minimal flow of the free abelian group on countably many generators. In Chapter 3, it is shown that the group $\Homeo(K)$ contains an open subgroup with a co-meager conjugacy class. In Chapter 4 (joint with Lukas Michel and Alex Scott) the main theorem is a Ramsey-type theorem about subsets of simplices which is motivated by a construction of Uspenskij. This is used along with Uspenskij's cosntruction to give a new proof of a theorem of Pestov--- that the group $\Homeo_+[0,1]$ of orientation-preserving homeomorphisms of the interval is extremely amenable. In Chapter 5 (joint with Forte Shinko), the main theorem is that the generic action of a countable free group on Cantor space generates a hyperfinite orbit-equivalence relation.