  For a commutative finite $\mathbb{Z}$-algebra, i.e., for a commutative ring
$R$ whose additive group is finitely generated, it is known that the group of
units of $R$ is finitely generated, as well. Our main results are algorithms to
compute generators and the structure of this group. This is achieved by
reducing the task first to the case of reduced rings, then to torsion-free
reduced rings, and finally to an order in a reduced ring. The simplified cases
are treated via a calculation of exponent lattices and various algorithms to
compute the minimal primes, primitive idempotents, and other basic objects. All
algorithms have been implemented and are available as a SageMath package.
Whenever possible, the time complexity of the described methods is tracked
carefully.
