First thing that comes to mind are circular dependencies.
Let's say we have a class that has a field referencing its child, and the child class has a field referencing its parent.
public class A
{
    public B Child;
}
public class B
{
    public A Parent;
}
public class Program
{
    private static void Main()
    {
        A a = new A();
        B b = new B();
        a.Child = b;
        b.Parent = a;
        string json = JsonConvert.SerializeObject(a);
    }
}
This would cause a run-time JsonSerializationException at JsonConvert.SerializeObject(a) with the following message: 
Self referencing loop detected for property 'Parent' with type 'A'. Path 'Child'.
To avoid this, JSON.NET provides an overload of SerializeObject to pass a settings object, where we can specifiy how to handle circular references.
JsonSerializerSettings settings = new JsonSerializerSettings
{
    PreserveReferencesHandling = PreserveReferencesHandling.None,
    ReferenceLoopHandling = ReferenceLoopHandling.Ignore
};
JsonConvert.SerializeObject(a, settings);
This way, the output json would completely ignore the circular reference from the child to the parent, and it would look like this:
{
    "Child": {}
}
JSON.NET also provides a way to handle them without losing information. We'd need to specifiy the option PreserveReferencesHandling.Objects in the settings.
JsonSerializerSettings settings = new JsonSerializerSettings
{
    PreserveReferencesHandling = PreserveReferencesHandling.Objects
};
The output JSON can only be interpreted by JSON.NET or other serializers compatible with the $id and $ref syntax, and it will look like this:
{
    "$id":"1",
    "Child": {
        "$id":"2",
        "Parent": {
            "$ref":"1"
        }
    }
}