  In this thesis we prove a variety of theorems on tournaments. A \emph{prime}
tournament is a tournament $G$ such that there is no $X \subseteq V(G)$, $1 <
|X| < |V(G)|$, such that for every vertex $v \in V(G) \minus X$, either $v \ra
x$ for all $x \in X$ or $x \ra v$ for all $x \in X$. First, we prove that given
a prime tournament $G$ which is not in one of three special families of
tournaments, for any prime subtournament $H$ of $G$ with $5 \le |V(H)| <
|V(G)|$ there exists a prime subtournament of $G$ with $|V(H)| + 1$ vertices
that has a subtournament isomorphic to $H$. We next prove that for any two
cyclic triangles $C$, $C^\prime$ in a prime tournament $G$, there is a sequence
of cyclic triangles $C_1,...,C_n$ such that $C_1 = C$, $C_n = C^\prime$, and
$C_i$ shares an edge with $C_{i+1}$ for all $1 \le i \le n-1$. Next, we
consider what we call \emph{matching tournaments}, tournaments whose vertices
can be ordered in a horizontal line so that every vertex is the head or tail of
at most one edge that points right-to-left. We determine the conditions under
which a tournament can have two different orderings satisfying the above
conditions. We also prove that there are infinitely many minimal tournaments
that are not matching tournaments. Finally, we consider the tournaments $K_n$
and $K_n^\ast$, which are obtained from the transitive tournament with $n$
vertices by reversing the edge from the second vertex to the last vertex and
from the first vertex to the second-to-last vertex, respectively. We prove a
structure theorem describing tournaments which exclude $K_n$ and $K_n^\ast$ as
subtournaments.
