The capability of the atomic force microscope (AFM) and the scanning tunnelling microscope (STM) to image surfaces with atomic resolution is still amazing us. Theory has been for a long time and in certain respects remains to the present day in debt to experiment. The aim of this review is to describe the development of theoretical models of the AFM, which serves both as an imaging tool and as a tool for manipulating atoms and molecules at surfaces. Models based on classical and quantum mechanical treatment of tip-sample interaction in the AFM are reviewed with emphasis on the explanations of atomic resolution that they provide. The attempts to understand atomic resolution in the STM on metal surfaces started with the local charge-density concept; however, even exact theories based on three-dimensional scattering theory, using different models for the tip and the sample, cannot provide an understanding of the large corrugation amplitudes of the tip height in the constant-current scanning mode in STM on densely packed metal surfaces. A recently developed dynamic theory of STM, regarding the tunnelling as an excited-state property of the interacting tip-plus-sample system, provides an insight into the physical background of atomic resolution in these cases. Other issues in the theory of STM and scanning tunnelling spectroscopy (STS) addressed relate to the equivalence of the current-density formulation of STM theory and the generalized Ehrenfest theorem, tunnelling via surface states and resonances, the mirror theorem of STS, image reversal in STM, many-particle effects in STM. With the prospect of converting the AFM and STM into tools for surface technology on the nanoscopic scale, theory will be challenged to suggest models going beyond the single-particle approaches and the adiabatic approximation.

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A combined experimental and theoretical study of the scanning-tunneling-miscroscope (STM) imaging properties of clean and alkali-metal covered Cu(110) and Au(110) surfaces is presented. The clean surfaces are imaged in the STM experiments as parallel strings of Cu or Au atoms, respectively, vVhich represent the close-packed rows in the topmost layer. For the (1 X 2) missing-row reconstructed Au(110) surface the corrugation amplitude of the reconstruction shows a maximum as a function of tip-sample distance. On the Cu(110) surface, which does not reconstruct spontaneously in its clean state, adsorbed alkali-metal atoms (K, Cs) induce a missing-row reconstruction with the missing substrate metal rows running along the densely packed [110]direction. On both surfaces, adatoms are located in the missingrow furrows. The alkali atoms are usually not visible in the STM picture but, rather, images are obtained typical for the reconstructed metal substrate. For certain tunneling conditions, image inversion is observed. K-covered Au(110)-(1X2) and Cu(110)-(1X2) surfaces exhibit distinct corrugation maxima similar to the clean Au(110)-(1X2) surface, if the tip-sample distance is varied. A theory of scanningtunneling microscopy is applied that accounts for a realistic treatment of the electronic structure of the sample surface. The tunnel current is evaluated using a Green-function technique. In the theory, adsorbed potassium atoms appear transparent on Cu(110) because they substitute for sample metal atoms and are embedded in the first layer of Cu atoms. Therefore, the 4s resonance is centered energetically well below the Fermi level and has only a small spectral weight at the Fermi level. The corrugation maximum and the image inversion are found to be a consequence of the tip-sample interaction.

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