This paper gives a unified probabilistic approach to the limit theory of
various types of supercritical branching processes. It is shown that
the key to the
convergence in probability of suitably normed branching processes is a law
of large numbers for a triangular array of independent random variables.
Criteria for convergence together with characterisations of the  norming
constants and limiting distributions follow from this property. The models
considered include the Galton--Watson and the general
age--dependent both in the simple and multitype case as well as in the
varying and random
environment settings. A martingale derived from a weakly convergent subsequence
is essential in the proofs.