The notions of completeness and compactness play important role in classical functional analysis. The main purpose of this paper is to generalize these notions to the setting of fuzzifying bornological linear spaces. At first, the concepts of fuzzifying Cauchy sequences and fuzzifying completeness are introduced and some interesting properties of them are studied. The relationships among fuzzifying completeness, separation axiom and fuzzifying bornological closed set are discussed. Then the notions of fuzzifying compactness and precompactness are presented, several properties of them are discussed. Particularly, it is demonstrated that a subset is fuzzifying bornological compact if and only if it is fuzzifying bornological precompact and bornological complete.