Let X be a compact Hausdorff space. A function algebra on X is a complex Banach subalgebra of C(X) which separates the points of X and contains the constants. Moreover, a function algebra on X is maximal if it is contained properly in no proper subalgebra of C(X). We mention that maximal function algebras are large enough to have a goodly amount of structure. In order that we be able to state the ideas and results simply let us assume that for each algebra A the underlying space X is so adjusted that A contains no non-trivial ideals of C(X). Generally if A is a maximal function algebra on X, then the topological dimension of X is at most one. The idea of this thesis is to extend the notion of maximal function algebra so that on the one hand features of maximal algebras would be retained, while on the other hand the topological dimension of the underlying space could be forced to be arbitrarily large. Thus our introduction of the notion of submaximal function algebra. We prove that all maximal algebras are submaximal. A submaximal, non-maximal algebra is A(Tn), the completion of the polynomials in n-complex variables on the unit n-torus in Cn. However, if A is submaximal on X, then each proper function algebra between A and C(X) is contained in a proper maximal function algebra on X. Moreover, we show by example that the converse to this last statement is false. If A is a submaximal function algebra on X, then every point in X has a compact neighborhood in X such that the algebra of restrictions of functions in A is dense in the continuous functions on the neighborhood. This is the (natural) analogue of the "pervasive" property of maximal function algebras. It turns out that maximal function algebras are antisymmetric, which means that they contain no non-constant real-valued functions. This is not true in general for submaximal function algebras. However, if we render the antisymmetric property in the following way, then it holds true for submaximal algebras: if the real-valued continuous functions f1,...,fn on X along with A together generate a dense subalgebra of C(X), then the continuous real-valued functions h1,...,hn on X and A together generate a dense subalgebra of C(X), provided only that each hj is sufficiently close to fj. In addition, we show that if A is submaximal on X, then there are always exist finitely many real-valued continuous functions on X which together with A generate a dense subalgebra of C(X). Finally we discuss tensor products of submaximal algebras. In particular, we prove that under certain restrictions, the tensor product of two submaximal algebras is submaximal.