This chapter introduces \(\textbf{d}\) -homogeneous functionsHomogeneous function and vector fieldsVector field in \(\mathbb {R}^n\) and studies their properties. In particular, a generalized Euler’s homogeneous function theoremHomogeneous function theorem is proven for linear dilationsLinear dilation; elements of “homogeneous arithmeticsHomogeneous arithmetics” are presented; regularity of \(\textbf{d}\) -homogeneous mappingsMapping is analyzed; homogeneous approximationsHomogeneous approximation and homogeneous extensionsHomogeneous extension of non-homogeneous mappingsMapping are discussed; a dilationDilation symmetry of tangent conesTangent cone of homogeneous setsSet is studied.