  We establish limit theorems involving weak convergence of multiple
generations of critical and supercritical branching processes. These results
arise naturally when dealing with the joint asymptotic behavior of functionals
defined in terms of several generations of such processes. Applications of our
main result include a functional central limit theorem (CLT), a Darling-Erd\"os
result, and an extremal process result. The limiting process for our functional
CLT is an infinite dimensional Brownian motion with sample paths in the
infinite product space $(C_0[0,1])^{\infty}$, with the product topology, or in
Banach subspaces of $(C_0[0,1])^{\infty}$ determined by norms related to the
distribution of the population size of the branching process. As an application
of this CLT we obtain a central limit theorem for ratios of weighted sums of
generations of a branching processes, and also to various maximums of these
generations. The Darling-Erd\"os result and the application to extremal
distributions also include infinite dimensional limit laws. Some branching
process examples where the CLT fails are also included.
