Traffic　information　can　be　used　for　real-time　traffic　management　and　long-term　transportation　planning　to　increase　traffic　efficiency　and　safety.　However,　data　containing　both　missing　and　deviating　values,　can　seriously　affect　the　accuracy　of　traffic　information,　even　leading　to　incorrect　results　in　traffic　data　analysis.　In　this　paper,　we　propose　a　novel　tensor-based　data　recovery　method　named　spatial-temporal　tensor　robust　principal　component　analysis　(ST-TRPCA)　to　recover　traffic　data　from　corrupted　and　incomplete　observations.　Specifically,　we　not　only　fully　account　for　the　spatial-temporal　properties　of　traffic　data　to　increase　the　data　recovery　accuracy,　but　also　utilize　tensor　factorization　and　its　low-dimensional　representation　to　improve　computational　efficiency.　The　extensive　experimental　results　performed　on　real-world　traffic　dataset　under　various　scenarios　show　that　ST-TRPCA　outperforms　other　state-of-the-art　methods　in　both　missing　data　recovery　and　anomaly　detection,　especially　when　the　traffic　data　are　severely　corrupted.