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A Costas Latin square of order \( n \) is a set of \( n \) disjoint Costas arrays of the same order. Costas Latin squares are studied here from both a construction and classification point of view. A complete classification is carried out up to order \( 27 \). In this range, we verify the conjecture that there is no Costas Latin square for any odd order \( n \geq 3 \). Various other related combinatorial structures are also considered, including near Costas Latin squares (which are certain packings of near Costas arrays) and Vatican Costas squares.