We review the present status of the singularity problem in gravitation theory. First we repeat the known results for Einstein's theory. After this, different other gravitation theories are considered under the following aspect: Which of the presumptions leading to singularities in Einstein's theory are not fulfilled, and can this prevent the singularities? In detail: scalar matter fields do not fulfil the positive energy condition, bimetric theories change the gravitational vacuum, theories with torsion and fourth order gravity add new geometric terms. It turns out that the answer is never simply no or yes, but requires new conditions, and new types of singularities occur. Otherwise there exist essential situations where the alternative theories satisfactorily prevent the singular behaviour known from Einstein's theory. Finally, we review on chaotic singularities.