Saturated model for Th(Z,+,-,0,1)? How an $\omega$-saturated model for the theory T=Th(Z,+,-,0,1) is made ?
Can you give me some concrete example? 
 A: The classic method to constructing an $\omega-$saturated model is by iterating compactness and constructing an elementary chain. However, this method is more abstract and does not allow you to interact with the elements of your structure (since the compactness theorem only tells you that a model exists, and so playing around with the elements is difficult since $(\mathbb{Z};0,1,+,-)$ has non-trivial structure). 
The facts I mention in my method above are overkill. However, this method allows us to "get a handle" on the elements in our model. Let $D$ be a non-principal ultrafilter over $I = \mathbb{N}$. Then, if we consider the ultrapower $\prod_D (\mathbb{Z}; +,-, 0,1)$, this structure is $\aleph_1 -$saturated $\implies$ $\omega-$saturated. The proof of this fact can be found in Chang and Keisler (the beginning of Chapter 6). Notice that we can think of an element of $\prod_D (\mathbb{Z}; +,-, 0,1)$ as a sequence of elements of $\mathbb{Z}$ indexed by the natural numbers and quotiented out by the ultrafilter. For instance $$[(1,1,1,1,1,...)]+[(-1,-1,-1,-1,-1,...)]=[(0,0,0,0,...)]= \mathbf{0}$$
and for $n \in \mathbb{Z}$
$$[(n,n,n,n,n,...)]+[(2,4,6,8,10,...)\neq \mathbf{0}$$
This method has it's advantages since it allows you to play around with (some of) the elements and provides intuition into how the elements of this structure interact. 
