"Rigid Cohomology over Laurent Series Fields" is a comprehensive monograph that develops a new theory of p-adic cohomology for varieties over Laurent series fields in positive characteristic. Based on Berthelot's theory of rigid cohomology, this work addresses essential properties of these cohomology theories, including finite dimensionality and cohomological descent. The authors also provide interpretations in the context of Monsky-Washnitzer cohomology and the overconvergent site of Le Stum. Furthermore, applications of this new theory to arithmetic questions, such as l-independence and the weight-monodromy conjecture, are discussed. The construction of these cohomology theories, which is analogous to Galois representations for varieties over local fields in mixed characteristic, fills a significant gap in the study of arithmetic cohomology theories over function fields. The work is aimed at anyone interested in the arithmetic of varieties over local fields of positive characteristic and offers extensive self-sufficiency through appendices to important background materials, which is particularly beneficial for graduate students.