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This thesis is devoted to algorithmic aspects of the implementation ofCartan's moving frame method to the problem of the equivalence of submanifoldsunder a Lie group action. We adopt a general definition of a moving frame as anequivariant map from the space of submanifolds to the group itself andintroduce two algorithms, which simplify the construction of such maps. Thefirst algorithm is applicable when the group factors as a product of twosubgroups $G=BA$, allowing us to use moving frames and differential invariantsfor the groups $A$ and $B$ in order to construct a moving frame anddifferential invariants for $G$. This approach produces the relations among theinvariants of $G$ and its subgroups. We use the groups of the projective, theaffine and the Euclidean transformations on the plane to illustrate thealgorithm. We also introduce a recursive algorithm, allowing, provided thegroup action satisfies certain conditions, to construct differential invariantsorder by order, at each step normalizing more and more of the group parameters,at the end obtaining a moving frame for the entire group. The development ofthis algorithm has been motivated by the applications of the moving framemethod to the problems of the equivalence and symmetry of polynomials underlinear changes of variables. In the complex or real case these problems can bereduced and, in theory, completely solved as the problem of the equivalence ofsubmanifolds. Its solution however involves algorithms based on the Gr\"obnerbasis computations, which due to their complexity, are not always feasible.Nevertheless, some interesting new results were obtained, such as aclassification of ternary cubics and their groups of symmetries, and thenecessary and sufficient conditions for a homogeneous polynomial in threevariables to be equivalent to $x^n+y^n+z^n.$

Inductive Approach to Cartan's Moving Frame Method with Applications to Classical Invariant Theory