The collision probability computation of space objects plays an important role in space situational awareness, particularly for conjunction assessment and collision avoidance. Early works mainly relied on Monte Carlo simulations to predict collision probabilities. Although such simulations are accurate when a large number of samples are used, these methods are perceived as computationally intensive, which limits their application in practice. To overcome this limitation, many approximation methods have been developed over the past three decades. This paper presents a comprehensive review of existing space-object collision probability computation methods. The advantages and limitations of different methods are analyzed and a systematic comparison is presented. Advice regarding how to select a suitable method for different short-term encounter scenarios is then provided. Additionally, potential future research avenues are discussed.