A standard theory of one-dimensional continued fractions is based on the sequential application of the so-called Gauss map and the comparison of the result to zero. The author proposes a generalization of this procedure to the case when one uses some other map, say \(A\), and an arbitrary stopping rule (e.g., the comparison to a given value \(\omega\) instead of zero as usual). For some specific choices of the map \(A\), the complete theory is obtained and further generalizations for infinite-dimensional vectors with integer coordinates are considered. In the finite-dimensional case, the author applies the generalized continued fractions to estimate certain trigonometric sums, while the infinite-dimensional generalizations are applied to estimate sums of Legendre symbols.