In this thesis, we wish to explore spreads of geometric spaces, and discuss some of their applications. In general, a spread of a geometry is a partition of the points of that geometry into pairwise disjoint substructures of some sort. We deal with three types of problems in this vein.

First, we consider the construction of translation planes, which are affine planes whose translations act transitively on its points. By a result of Bruck and Bose, constructing translation planes is equivalent to constructing spreads of odd-dimensional projective spaces. A good deal of work has been done in constructing spreads of $/cal[PG](3,q);$ however not much is known in higher dimensions. We construct some new spreads of $/cal[PG](5,q),$ and discuss some properties of the associated translation planes.

Second, we look at a problem related to spreads of the five-dimensional elliptic quadric. Spreads of this quadric have been used to construct semi-partial geometries and perfect codes in graphs. We give a geometric presentation of some classical constructions of packings, or partitions of the lines of the quadric into pair-wise disjoint spreads, and use this presentation to extend some results of Brouwer and Wilbrink. In particular, we make a connection between packings of Hermitian unitals and packings of the elliptic quadric.

Finally, we consider some questions about parabolic Buekenhout unitals, using the hyperspace model of Bruck and Bose. Unitals are generalizations of some classical curves over the complex field. In 1976, Buekenhout gave a general method for constructing unitals in any two-dimensional translation plane. The study of unitals has been enhanced by some recently discovered applications to coding theory; in particular, unitals can be used to give an interesting class of algebraic geometry codes. Also, spreads of unitals have been used to investigate the code associated with the Desarguesian plane.

Here, we discuss the existence of spreads and packings of these unitals, along with some related design theoretic questions. Then, starting with these unitals in the Desarguesian plane, we give a construction for a family of unitals in the Hall planes. These unitals are the first known examples of unitals embedded in any translation plane which do not arise from either of Buekenhout's constructions.