I. INTRODUCTION During the last few decades possible existence of Yukawa-type corrections to Newtonian gravitational law has attracted considerable attention [1]. In the middle of eighties the problem of the so-called fifth force was widely discussed. Finally, no deviations from the predictions of Newtonian gravity have been found. However, after some period of time interest in hypothetical corrections to Newton’s law at short separations was rekindled by numerous predictions of high energy physics beyond the Standard Model. On the one hand, many unification schemes predicted the existence of massless and light bosons such as arion [2], scalar axion [3], graviphoton [4], dilaton [5], goldstino [6], moduli [7], etc. The exchange of light bosons between two atoms belonging to different macrobodies generates an effective Yukawa-type force at short range depending on the mass of the particle. On the other hand, the Yukawa-type correction to Newton’s law at submillimeter separations was predicted by multidimensional unification schemes, where additional spatial dimensions are compactified at relatively low energy of the order of 1 TeV [8–10]. In such schemes, at separations much larger than the size of a compact manifold, the gravitational potential is given by the sum of Newton’s and Yukawa-type terms [11, 12]. Keeping in mind that at separations below 10 µ m the Newton’s law was poorly tested experimentally, the above theoretical predictions generated considerable public excitement. The key question is whether the Yukawa-type corrections to Newtonian gravity exist in Nature and, if they do, how strong they are and what is their interaction range. The strongest constraints on the parameters of these corrections in the interaction range larger than 4 . 7 µ m were obtained from the E¨ otvos [13, 14] and Cavendish-type [15–18] experi- ments. At shorter interaction range, however, the gravitational experiments do not lead to competitive constraints because the gravitational force becomes too weak. As to very short separations, the hypothetical interactions of Yukawa-type should be considered in a background of the van der Waals and Casimir forces. The possibility of obtaining constraints on the predicted forces of Yukawa and power-type from the measurements of the Casimir force was predicted in Refs. [19, 20], respectively. The Casimir force originates from zero-point and thermal fluctuations of the electromagnetic field. It acts at separations of the order of 1 µ m between the surfaces of uncharged material bodies (see Ref. [21] for a recent overview on the subject). The modern stage in the measurement of 2