Electro-static or electro-dynamic fields generated by micro- or nano-electrodes are today widely used in many different fields, such as micro- and nanogripping, "lab on a chip", and "lab in a cell", for the purpose of manipulation, separation, or analysis of micron-sized particles, cells, or single molecules. Commonly numerical simulations and Electrostatic Force Microscopy (EFM) methods, based on Atomic Force Microscopy (AFM), are used to study and predict the electrostatic force fields above produced structures. However, this measurement methods have several disadvantages. These disadvantages range from simple limitations (e.g. it is not possible to measure in three dimensions or expensive new equipment is needed) to much more serious disadvantages (e.g. missing compensation for known AFM problems, which results in wrong measurements). In this thesis we propose a new EFM method based on simple static force distance curves, which allows measuring accurately, and simultaneously the topography, vertical electrostatic force field and attraction forces. The method is numerical simulated in 3D to study in detail, with simulations and measurements, electrostatic force fields above nanoelectrodes. To do this, we show for the first time the fabrication of interdigitated nanoelectrodes with pitches down to 50 nm with gas assisted focused ion beam milling. We present as well for the first time the fabrication of Pt coated AFM tips with radii between 50 and 600 nm, together with their calibration. These tips not only close the gap between conventional tips with tip radii of about 10 nm and cantilevers with attached spheres. They show as well high mechanical stability, which solves a common problem in EFM, the mechanic stability of the tip. The shape of the tip greatly influences the measurement. We demonstrate this using numerical simulations, and show that already small tip discrepancies can change the measured force up to 50 %. Moreover, we proved that changes in the relative humidity result in electrostatic force changes of up to 45 %. We discovered that the resolution of EFM measurements on nanoelectrodes can be enhanced by using a tip radius of 2 to 2.5 times the pitch of the measured interdigitated nanoelectrode. Based on this and its influence on topography measurements, which are usually made before, in between or at the end of EFM measurements, we give hints for measurements with different tip radii. Beside these important general improvements, discoveries and hints for EFM, we show as well some applications of our new EFM method and the before obtained results. We show that our new method can distinguish the force field caused by trapped charges in SiO2 from the force field caused by nanoelectrodes below it. This enables to study the influence of charge trapping in future semiconductor chips, which use charge trapping to store information. Furthermore, we used our new EFM method and some gripping experiments to study electrostatic micro-and nanogripping. This experiments lead to some propositions for improvements in electrostatic micro-and nanogripping. Such as a new gripper design with gripping object size independent centering effect, which is studied using 3D numerical simulations.