The vapour pressure of both liquid and solid thin films is expected, on thermodynamic grounds, to differ from that of the bulk material. An investigation of this effect has been carried out using an electron microscope and the films so obtained in these experiments took the form of small discrete islands. Observations showed that over a certain temperature range these islands assumed spherical shapes By allowing the spherical drops to evaporate in the electron microscope and photographing them at regular intervals, the radius of a given drop could be determined as a function of time, It was found that at large radii this function was linear: but for radii smaller than 100 Å the slope of the function increased considerably. It is shown in the thesis that the vapour pressure of the drop is proportional to this slope. By using the above method a quantitative test of the equation derived by Lord Kelvin, which relates vapour pressure to surface curvature, was carried out for drops with curvatures much greater than was hitherto possible. These experiments, which were performed with both liquid and solid drops, gave results which were in good agreement with the Kelvin equation. A review of earlier experiments undertaken to test the Kelvin equation is also given in this thesis.