The famous ham-sandwich theorem says that d “nice” sets in d
−dimensional space 
may be simultaneously split in half by a single (d −
1)−dimensional hyperplane. This 
is an example of geometric partitioning, and there is a growing number
of interesting 
and useful statements of this kind. At the same time there are negative
results stating 
that certain kinds of partitions are, in general, impossible. This
subject has a direct 
connection to computer science through the study of algorithms for
geometric parti- 
tioning in the discrete context of points in R^d . I will describe a
variety of positive and 
negative results of this kind, and try to convince you of the interest
and importance 
of this topic in CS by showing some recent partitioning algorithms, and
many open 
questions. The details should be intelligible to non-specialists.