The author extends the theory of almost coherent modules that was introduced in Almost Ring Theory, by Gabber and Ramero (2003). Then he globalizes it by developing a new theory of almost coherent sheaves on schemes and on a class of ‘nice’ formal schemes. He shows that these sheaves satisfy many properties similar to usual coherent sheaves, i.e., the amost proper mapping theorem, the formal GAGA, etc. He also constructs an almost version of the Grothendieck twisted image functor \(f!\) and verifies its properties. Lastly, he studies sheaves of \(p\)-adic nearby cycles on admissible formal models of rigid-analytic varieties and shows that these sheaves provide examples of almost coherent sheaves. This gives a new proof of the finiteness result for étale cohomology of proper rigid-analytic varieties obtained before in Peter Scholze's \(p\)-adic Hodge theory for rigid-analytic varieties (2013).