Sobolev Spaces are mathematical constructs used in functional analysis and partial differential equations. They provide a framework for studying functions that have certain smoothness properties, allowing for the inclusion of functions that may not be differentiable in the traditional sense. These spaces are denoted as W^{k,p} where k indicates the number of derivatives and p represents the integrability condition. The significance of Sobolev Spaces lies in their ability to handle weak derivatives, which are essential for solving various problems in mathematics and physics. They enable the formulation of variational problems and the study of boundary value problems, making them crucial in the analysis of PDEs and functional analysis.