A shortest-path algorithm finds a path containing the minimal cost between two vertices in a graph. A plethora of shortest-path algorithms is studied in the literature that span across multiple disciplines. This paper presents a survey of shortest-path algorithms based on a taxonomy that is introduced in the paper. One dimension of this taxonomy is the various flavors of the shortest-path problem. There is no one general algorithm that is capable of solving all variants of the shortest-path problem due to the space and time complexities associated with each algorithm. Other important dimensions of the taxonomy include whether the shortest-path algorithm operates over a static or a dynamic graph, whether the shortest-path algorithm produces exact or approximate answers, and whether the objective of the shortest-path algorithm is to achieve time-dependence or is to only be goal directed. This survey studies and classifies shortest-path algorithms according to the proposed taxonomy. The survey also presents the challenges and proposed solutions associated with each category in the taxonomy.