  The properties of gauge-invariant composite operators and their correlation
functions in N=4 SYM are discussed in the analytic superspace formalism. A
complete classification of the different types of operators in the theory is
given. Operators can be either protected or unprotected according to whether
they do not or do have anomalous dimensions, and the analytic superspace
formalism allows one to identify which type a given operator is in a
straightforward manner. A simple discussion is given of the behaviour of
reducible multiplets at threshold. It is pointed out that there is a class of
``semi-protected'' operators which do not have anomalous dimensions but which
do not necessarily have non-renormalised three-point functions when the other
two operators in the correlator are protected, although two-point functions of
such operators are non-renormalised. A complete discussion of superconformal
invariants in analytic superspace is given. The paper includes a modified
discussion of the transformation rules of analytic superfields which clarifies
the $U(1)_Y$ properties of operators and correlation functions and, in
particular, explicit examples are given of three-point correlation functions
which violate this symmetry. A tensor, $\cE$, invariant under $SL(n|m)$ but not
under $GL(n|m)$, is introduced and used in the discussion of $U(1)_Y$ and in
the construction of invariants.
