Constructing groups of a given order

Group theory is an area of algebra. It originated from the description of the symmetries of polynomials. Nowadays it has many applications in different areas of mathematics, physics and computer science. The number of elements in a group -- its order -- can be finite or infinite. The finite groups have long intrigued algebraists, because they formalise the concept of 'symmetry' and have applications ranging from the insolubility of the quintic equation to crystallography. A huge amount is known about finite groups, including a complete classification of the finite simple groups: these are the building blocks of all finite groups. However, there is a big difference between knowing about the individual building blocks and knowing about everything that can be build with them. This talk gives a survey on the history and the modern methods in the classification of all groups of a given order.