In a series of papers published between 1965 and 1972,

W.M. Schmidt generalized Roth's Theorem from 1955,

which states that for every real algebraic number x

the Diophantine inequality |x - p/q| < 1/q^e, e > 2,

has only finitely many integral solutions p,q.

The Subspace Theorem is the multidimensional generalization

of this result; it essentially tells us something about the number

of solutions satisfying some linear form under certain conditions.

Therefore, it can be used to derive upper bounds for the number of

solutions of Diophantine equations algorithmically.

In this talk we also present recent applications of this

deep theorem and show how it can be used to handle certain classes

of Diophantine equations, which are of exponential-polynomial type.