"In his 1950 Congress address, Hodge reported on the topological and differential-geometric methods in studying algebraic varieties and complex manifolds which had been initiated by Lefschetz and developed by Hodge himself. He raised there many problems, and most of them were settled in 1950's by extensive works due to Kodaira and others. One notable exception to this is the so-called Hodge Conjecture which, if true, will give a characterization of cohomology classes of algebraic cycles on a nonsingular projective variety, generalizing the Lefschetz criterion for the case of divisors. This conjecture has an arithmetic flavour, as is common to most problems concerning algebraic cycles, which makes the problem interesting and difficult at the same time."