  We report on an ion-optical system that serves as a microscope for ultracold
ground state and Rydberg atoms. The system is designed to achieve a
magnification of up to 1000 and a spatial resolution in the 100 nm range,
thereby surpassing many standard imaging techniques for cold atoms. The
microscope consists of four electrostatic lenses and a microchannel plate in
conjunction with a delay line detector in order to achieve single particle
sensitivity with high temporal and spatial resolution. We describe the design
process of the microscope including ion-optical simulations of the imaging
system and characterize aberrations and the resolution limit. Furthermore, we
present the experimental realization of the microscope in a cold atom setup and
investigate its performance by patterned ionization with a structure size down
to 2.7 {\mu}m. The microscope meets the requirements for studying various
many-body effects, ranging from correlations in cold quantum gases up to
Rydberg molecule formation.
