  The $d$-dimensional hypercube is the $d$-dimensional generalisation of a
square $(d=2)$ and a cube $(d=3)$. Formally, it is the graph with vertex set
$\{0,1\}^d$ and edges connecting any two vertices that differ in exactly one
coordinate. These vertices represent the corners of the hypercube, while the
edges establish connections between these corners. It is well-known that for $d
\geq 2$ the $d$-dimensional hypercube contains a Hamilton cycle, which is a
path that visits every vertex exactly once and returns to the starting vertex.
  In this thesis we address the analogous problem for the $3$-uniform cube
hypergraph, a $3$-uniform analogue of the hypercube. In contrast to the
ordinary hypercube, the vertex set is now $\{0,1,2\}^d$ and edges consist of
three instead of two vertices. We will examine the $3$-uniform cube hypergraph
and consider a number of different definitions of paths and cycles. Our
strongest result concerns loose paths: for simple parity reasons, the
$3$-uniform cube hypergraph can never admit a loose Hamilton cycle in any
dimension, but we determine for which dimensions a loose Hamilton path exists.
