  In this thesis we develop the cohomology of diagrams of algebras and then
apply this to the cases of the $\lambda$-rings and the $\Psi$-rings. A diagram
of algebras is a functor from a small category to some category of algebras.
For an appropriate category of algebras we get a diagram of groups, a diagram
of Lie algebras, a diagram of commutative rings, etc.
  We define the cohomology of diagrams of algebras using comonads. The
cohomology of diagrams of algebras classifies extensions in the category of
functors. Our main result is that there is a spectral sequence connecting the
cohomology of the diagram of algebras to the cohomology of the members of the
diagram.
  $\Psi$-rings can be thought of as functors from the category with one object
associated to the multiplicative monoid of the natural numbers to the category
of commutative rings. So we can apply the theory we developed for the diagrams
of algebras to the case of $\Psi$-rings. Our main result tells us that there is
a spectral sequence connecting the cohomology of the $\Psi$-ring to the
Andr\'{e}-Quillen cohomology of the underlying commutative ring.
  The main example of a $\lambda$-ring or a $\Psi$-ring is the $K$-theory of a
topological space. We look at the example of the $K$-theory of spheres and use
its cohomology to give a proof of the classical result of Adams. We show that
there are natural transformations connecting the cohomology of the $K$-theory
of spheres to the homotopy groups of spheres. There is a very close connection
between the cohomology of the $K$-theory of the $4n$-dimensional spheres and
the homotopy groups of the $(4n-1)$-dimensional spheres.
