In the classical iteration theory we say that for a given polynomial $f$a point $z_0\in\C$ belongs to the Julia set if the sequence ofiterates $(f^n)$ is not normal in any neighbourhood of $z_0$. In thispaper, we look at the set of non-normality of $(F_n)$,$F_n:=f_n\circ\cdots\circ f_1$, where $(f_n)$ is a given sequence ofpolynomialsof degree at least two. If we can find a hyperbolic domain $M$ whichis invariant under all $f_n$, $n\in\N$, $\infty\in M$ and$F_n\to\infty\ (n\to\infty)$ locally uniformly in $M$, then weask whether these sets of non-normality,which we will also call Julia sets, have properties which we know from theclassical case. We show that the Julia set is self-similar. Furthermore,the Julia set is perfect or finite. The finite case may actually occur.We will also give some sufficient conditions for the Julia set being perfect.In the last section we give some examples of sequences of polynomials (whereno domain $M$ exists) which have a pathological behaviour in contrast to theclassical case.