A group described in terms of a group presentation
can be difficult to understand. In general, even determining
if such a group is trivial is not algorithmically decidable.
However, it is easy to define group homomorphisms whose domain
has a known presentation, and these homomorphisms can be used
to study the domain and its elements. In this talk, we investigate
a property of groups called residual finiteness, which guarantees
the existence of especially useful homomorphisms. We give examples
of presentations which define groups that are or are not residually
finite. We then explore characterizations of residual finiteness,
stronger and weaker conditions, and methods of constructing new
residually finite groups from old.