This paper considers compactness in a ditopological setting. After a brief introduction, Section 2 is devoted to compact and cocompact spaces. Results include preservation under surjective (co) continuous difunctions, and analogues of the Mrowka Charcterization and Tychonoff Product Theorem. Stability and costability are discussed in Section 3. Here generalizations of several results concerning separation are presented, characterizations of the compact and cocompact elements of the texturing given under Suitable conditions and the preservation of stability and costability under surjective bicontinuous difunctions established. Finally Section 4 considers dicompactness and cumulates with a proof of the Tychonoff Product Theorem for this very important class of ditopological texture spaces.