As the data handling tasks required of space-borne astronomical instrumentation grow in size and complexity, more and more autonomy will be required of on-board control and analysis software. One area in which this is especially relevant is in the area of target acquisition. One simple method of target acquisition involves using on-board software to construct and downlink map images of areas of the sky as seen by on-board detectors. These maps are analyzed on the ground, and the on-board instrumentation package is then commanded from the ground to point in the proper direction. While this method is workable, it is far from optimal, since increasingly expensive real-time downlink data paths and uplink command paths are required. A more desirable method of target acquisition requires no real-time data or command links to or from the ground, and is carried out completely autonomously by on-board digital image processing and spacecraft pointing software. Such a software package was designed and implemented as part of the Flight Software for the High Resolution Spectrograph (HRS) for the Space Telescope (ST). This paper discusses the generic target acquisition process by using the HRS process as an example. Precisely because of the autonomous actions of the software, the ground-testing of autonomous on-board target acquisition software prior to launch is a crucial activity. In the absence of ST end-to-end target acquisition test facilities, a realistic test bed was nonetheless devised in order to test all features of the HRS target acquisition flight software. This paper briefly discusses some of the actual testing which has taken place. The HRS contains detectors which are 1-dimensional and provide a linear uniform lx500 pixel format, but sophisticated 2nd generation ST instruments are planned which contain 2-dimensional detectors providing formats of 2048x2048 pixels. Some implications of using 2-D detectors for target acquisition are briefly examined.