  The first section starts with the basic definitions following mainly the
notations of the book written by E. Kushilevitz and N. Nisan. At the end of the
first section I examine tree-balancing.
  In the second section I summarize the well-known lower bound methods and
prove the exact complexity of certain functions.
  In the first part of the third section I introduce the random complexity and
prove the basic lemmas about it. In the second part I prove a better lower
bound for the complexity of all random functions. In the third part I introduce
and compare several upper bounds for the complexity of the identity function.
  In the fourth section I examine the well-known Direct-sum conjecture. I
introduce a different model of computation then prove that it is the same as
the original one up to a constant factor. This new model is used to bound the
Amortized Time Complexity of a function by the number of the leaves of its
protocol-tree. After this I examine the Direct-sum problem in case of Partial
Information and in the Random case.
  In the last section I introduce the well-known hierarchy classes, the
reducibility and the completeness of series of functions. Then I define the
class PSPACE and Oracles in the communication complexity model and prove some
basic claims about them.
