Two ways of describing a group are considered.
1. A group is finite-automaton presentable
if its elements can be represented by strings over a finite alphabet,
in such a way that the set of representing strings and
the group operation can be recognized by finite automata.
2. An infinite f.g. group is quasi-finitely axiomatizable
if there is a description consisting of a single first-order sentence,
together with the information that the group is finitely generated.
In the first part of the paper we survey examples of FA-presentable
groups, but also discuss theorems restricting this class. In the second part,
we give examples of quasi-finitely axiomatizable groups,
consider the algebraic content of the notion,
and compare it to the notion of a group which is a prime model.
We also show that if a structure is bi-interpretable in parameters
with the ring of integers, then it is prime and quasi-finitely
axiomatizable.