  This work provides a brief introduction to continuous-state branching
processes (CB-processes) and continuous-state branching processes with
immigration (CBI-processes) accessible to graduate students with reasonable
background in probability theory and stochastic processes. In particular, we
give a quick development of the stochastic equations of the processes and some
immediate applications. The proofs given here are more elementary than those
appearing in the literature before. We have made them readable without
requiring too much preliminary knowledge on branching processes and stochastic
analysis. In Section 1, we review some properties of Laplace transforms of
finite measures on the positive half line. In Section 2, a construction of
CB-processes is given as rescaling limits of Galton--Watson branching
processes. This approach also gives the physical interpretation of the
CB-processes. Some basic properties of the processes are developed in Section
3. The Laplace transforms of some positive integral functionals are calculated
explicitly in Section 4. In Section 5, the CBI-processes are constructed as
rescaling limits of Galton--Watson branching processes with immigration. In
Section 6, we present reconstructions of the CB- and CBI-processes by Poisson
random measures determined by entrance laws, which reveal the structures of the
trajectories of the processes. Several equivalent formulations of martingale
problems for CBI-processes are given in Section 7. From those we derive the
stochastic equations of the processes in Section 8. Using the stochastic
equations, some characterizations of local and global maximal jumps of the CB-
and CBI-processes are given in Section 9. In Section 10, we prove the strong
Feller property and the exponential ergodicity of the CBI-process under
suitable conditions using a coupling based on one of the stochastic equations.
