This paper explores the properties of homogeneous systems in a geometric, coordinate-free setting. A key contribution is a result that links the regularity properties of a homogeneous function to its degree of homogeneity and the local behavior of the dilation near the origin. This result extends previous work on homogeneous systems to a geometric framework. The main application discussed is the finite-time stability of homogeneous systems, where it is shown that a homogeneous system is finite-time stable if and only if it is asymptotically stable and has a negative degree of homogeneity. The paper also demonstrates that homogeneity leads to stronger properties for finite-time stable systems. Homogeneity is defined in relation to a scaling operation or dilation, which is an action of the multiplicative group of positive real numbers on the state space. The standard dilation, where each coordinate is scaled by a positive power of the dilation parameter, is a special case. Recent work has extended the study to dilations where each coordinate is scaled by a different power of the dilation parameter. Asymptotic stability of the origin implies global asymptotic stability and the existence of a \(C^1\) Lyapunov function that is also homogeneous with respect to the same dilation. The stability of a homogeneous system is determined by its restriction to certain invariant sets, similar to linear systems. An important application of homogeneity is deducing the stability of a nonlinear system from the stability of a homogeneous approximation. If a vector field can be written as the sum of several vector fields, each homogeneous with respect to a fixed dilation, then asymptotic stability of the lowest degree vector field implies local asymptotic stability of the original vector field. This generalizes Lyapunov's first method of stability analysis, where the Taylor series expansion is used to write a given analytic vector field as a sum of homogeneous vector fields. The paper also discusses homogeneous stabilization and connections between stabilizability and homogeneous feedback stabilization.This paper explores the properties of homogeneous systems in a geometric, coordinate-free setting. A key contribution is a result that links the regularity properties of a homogeneous function to its degree of homogeneity and the local behavior of the dilation near the origin. This result extends previous work on homogeneous systems to a geometric framework. The main application discussed is the finite-time stability of homogeneous systems, where it is shown that a homogeneous system is finite-time stable if and only if it is asymptotically stable and has a negative degree of homogeneity. The paper also demonstrates that homogeneity leads to stronger properties for finite-time stable systems. Homogeneity is defined in relation to a scaling operation or dilation, which is an action of the multiplicative group of positive real numbers on the state space. The standard dilation, where each coordinate is scaled by a positive power of the dilation parameter, is a special case. Recent work has extended the study to dilations where each coordinate is scaled by a different power of the dilation parameter. Asymptotic stability of the origin implies global asymptotic stability and the existence of a \(C^1\) Lyapunov function that is also homogeneous with respect to the same dilation. The stability of a homogeneous system is determined by its restriction to certain invariant sets, similar to linear systems. An important application of homogeneity is deducing the stability of a nonlinear system from the stability of a homogeneous approximation. If a vector field can be written as the sum of several vector fields, each homogeneous with respect to a fixed dilation, then asymptotic stability of the lowest degree vector field implies local asymptotic stability of the original vector field. This generalizes Lyapunov's first method of stability analysis, where the Taylor series expansion is used to write a given analytic vector field as a sum of homogeneous vector fields. The paper also discusses homogeneous stabilization and connections between stabilizability and homogeneous feedback stabilization.