  Many of the numerous automaton models proposed in the literature can be
regarded as a finite automaton equipped with an additional storage mechanism.
In this thesis, we focus on two such models, namely the finite automata over
groups and the homing vector automata.
  A finite automaton over a group $ G $ is a nondeterministic finite automaton
equipped with a register that holds an element of the group $ G $. The register
is initialized to the identity element of the group and a computation is
successful if the register is equal to the identity element at the end of the
computation after being multiplied with a group element at every step. We
investigate the language recognition power of finite automata over integer and
rational matrix groups and reveal new relationships between the language
classes corresponding to these models. We examine the effect of various
parameters on the language recognition power. We establish a link between the
decision problems of matrix semigroups and the corresponding automata. We
present some new results about valence pushdown automata and context-free
valence grammars.
  We also propose the new homing vector automaton model, which is a finite
automaton equipped with a vector that can be multiplied with a matrix at each
step. The vector can be checked for equivalence to the initial vector and the
acceptance criterion is ending up in an accept state with the value of the
vector being equal to the initial vector. We examine the effect of various
restrictions on the model by confining the matrices to a particular set and
allowing the equivalence test only at the end of the computation. We define the
different variants of the model and compare their language recognition power
with that of the classical models.
