The fuzzy set theory introduced by Zadeh, 1965 is considered as one of the powerful tools in handling the uncertainty and vague concepts in a more practical way with great progress in every scientific research area. Fuzzy sets (FSs) have number of applications in different research fields such as engineering, computer sciences, arts, humanities, life sciences, mathematics, and health sciences and moreover, it has been extended to number of new types. Among all the extensions, intuitionistic fuzzy sets (IFSs) introduced by Atanassov, 1986 broadens the field of fuzzy sets in a reliable way by adding the third component called as Hesitancy index (π) along with the membership (µ) and non-membership (v) functions. For the decision-making problems, MCDM (Multi-criteria decision making) is considered as one of the appropriate weapons in finding the best alternative among the different criterion. The aim of present review paper is to provide a detailed analysis on the multi-criteria decision making techniques under the fuzzy and intuitionistic fuzzy set from the year 2011 to 2015. We also described how fast the role of fuzzy theory increases in solving the MCDM (multicriteria decision making) issues. In addition to this, we also systematically conduct a review on the different applications and methodologies of MCDM techniques such as AHP, PROMETHEE, ELECTRE, TOPSIS, VIKOR, and ANP etc. [ABSTRACT FROM AUTHOR]