The present review paper intends to briefly summarize a broad range of research on various stability problems of compressed beams. It is noted that multiple analysis approaches have been used, many are closed-form analytical, while others are classified as semi-analytical or simply numerical methods. This study overviews some of the most important theories and solutions for beam buckling from the 18th century up to date, bringing together the findings from several scientific perspectives. The emphasis is put on the recent novel results.