This paper addresses a decision problem highlighted by Grigorchuk, Nekrashevich, and Sushchanskiȋ, namely the finiteness problem for automaton (semi)groups. For semigroups, we give an effective sufficient but not necessary condition for finiteness and, for groups, an effective necessary but not sufficient condition. The efficiency of the new criteria is demonstrated by testing all Mealy automata with small stateset and alphabet. Finally, for groups, we provide a necessary and sufficient condition that does not directly lead to a decision procedure.Two GAP packages are dedicated to automaton (semi)groups: FR by Bartholdi and automgrp by Muntyan and Savchuk [4,13,23]. Both include specific (in)finiteness tests. Besides the three results above-mentioned, all that was known up to now about the finiteness question for automaton groups happened to be somehow summarized in the documentation for FR:The order of [an automaton] group is computed as follows: if all [the states have growth degree at most 0], then enumeration will succeed in computing the order. If the action of the group is primitive, and it comes from a bireversible automaton, then the Thompson-Wielandt theorem is tested against [. . .] see [11, Prop. 2.1.1]. Then, FR attempts to find whether the group is level-transitive (in which case it would be infinite). Finally, it attempts to enumerate the group's elements, testing at the same time whether these elements have infinite order.Needless to say, none except the first few steps are guaranteed to succeed.