  In a properly edge colored graph, a subgraph using every color at most once
is called rainbow. In this thesis, we study rainbow cycles and paths in proper
edge colorings of complete graphs, and we prove that in every proper edge
coloring of K_n, there is a rainbow path on (3/4-o(1))n vertices, improving on
the previously best bound of (2n+1)/3 from Gyarfas and Mhalla. Similarly, a
k-rainbow path in a proper edge coloring of K_n is a path using no color more
than k times. We prove that in every proper edge coloring of K_n, there is a
k-rainbow path on (1-2/(k+1)!)n vertices.
