The purpose of this thesis is to introduce the notion of (3,4)-fuzzy sets. (3,4)-fuzzy sets are compared to intuitionistic fuzzy sets, Pythagorean fuzzy sets, and Fermatean fuzzy sets. The focus is on the complement of (3,4)-fuzzy sets. We develop some of the fundamental operations of the (3,4)-fuzzy sets. (3,4)-fuzzy sets can deal with more uncertain situations than other types of fuzzy sets because they have a wider range of describing membership grades. A score function and an accuracy function for ranking (3,4)-fuzzy sets are defined. The idea of (3,4)-fuzzy relation has been established with some properties. The decision-making strategy for child placement is provided using a (3,4)-fuzzy relation to determine the suitability of fathers to applicants based on their blood types. Moreover, four weighted aggregated operators; namely, (3,4)-fuzzy weighted average ((3,4)-FWA), (3,4)-fuzzy weighted geometric ((3,4)-FWG), (3, 4)- fuzzy weighted power average((3,4)-FWPA), and (3,4)-fuzzy weighted power geometric ((3,4)-FWPG) over (3,4)-fuzzy sets are constructed. These operators are used to determine the top-ranked football striker and illustrate how we can select the best choice by comparing the aggregate outputs using score values. Also, the concept of (3,4)-fuzzy topological space is introduced. We define and characterize (3,4)-fuzzy continuity of a map defined between (3, 4)-fuzzy topological spaces. Then, the concept of (3,4)-fuzzy points is introduced and different kinds of separation axioms in (3, 4)-fuzzy topological space are studied. Finally, (3,4)-fuzzy compactness and (3,4)-fuzzy connectedness are constructed.