This paper studies homogeneous systems in a geometric, coordinate-free setting. A key contribution is a result linking the regularity properties of a homogeneous function to its degree of homogeneity and the local behavior of the dilation near the origin. This result allows extending previous results on homogeneous systems to the geometric framework. As an application, the paper considers finite-time stability of homogeneous systems. The main result is 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 shows that homogeneity leads to stronger properties for finite-time stable systems. Homogeneity is defined with respect to a dilation, which is an action of the multiplicative group of positive real numbers on the state space. The standard dilation is Δ_λ(x) = λx, where λ > 0 and x ∈ R^n. Homogeneity with respect to the standard dilation is one of the two axioms for linearity, the other being additivity. Many familiar properties of linear systems follow from homogeneity alone. Recent work has extended homogeneity to dilations of the form Δ_λ(x) = (λ^{r1}x1, ..., λ^{rn}xn), where r_i are positive real numbers. The standard dilation is a special case with r1 = ... = rn = 1. Many recent results on homogeneous systems are generalizations of familiar properties of linear systems. For example, asymptotic stability of the origin implies global asymptotic stability and the existence of a C^1 Lyapunov function for homogeneous systems. Stability of a homogeneous system is determined by its restriction to certain invariant sets. An important application of homogeneity is in deducing the stability of a nonlinear system from the stability of a homogeneous approximation. A general result states that if a vector field can be written as the sum of several homogeneous vector fields, then asymptotic stability of the lowest degree vector field implies local asymptotic stability of the original vector field. This is similar to 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. Stability of the given vector field is deduced from the stability of the lowest degree vector field in the sum, which is the linearization of the given vector field. Homogeneous stabilization of homogeneous systems is considered in [K4, K6, SA1], while connections between stabilizability and homogeneous feedback stabilization are explored in [H7, SA3].This paper studies homogeneous systems in a geometric, coordinate-free setting. A key contribution is a result linking the regularity properties of a homogeneous function to its degree of homogeneity and the local behavior of the dilation near the origin. This result allows extending previous results on homogeneous systems to the geometric framework. As an application, the paper considers finite-time stability of homogeneous systems. The main result is 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 shows that homogeneity leads to stronger properties for finite-time stable systems. Homogeneity is defined with respect to a dilation, which is an action of the multiplicative group of positive real numbers on the state space. The standard dilation is Δ_λ(x) = λx, where λ > 0 and x ∈ R^n. Homogeneity with respect to the standard dilation is one of the two axioms for linearity, the other being additivity. Many familiar properties of linear systems follow from homogeneity alone. Recent work has extended homogeneity to dilations of the form Δ_λ(x) = (λ^{r1}x1, ..., λ^{rn}xn), where r_i are positive real numbers. The standard dilation is a special case with r1 = ... = rn = 1. Many recent results on homogeneous systems are generalizations of familiar properties of linear systems. For example, asymptotic stability of the origin implies global asymptotic stability and the existence of a C^1 Lyapunov function for homogeneous systems. Stability of a homogeneous system is determined by its restriction to certain invariant sets. An important application of homogeneity is in deducing the stability of a nonlinear system from the stability of a homogeneous approximation. A general result states that if a vector field can be written as the sum of several homogeneous vector fields, then asymptotic stability of the lowest degree vector field implies local asymptotic stability of the original vector field. This is similar to 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. Stability of the given vector field is deduced from the stability of the lowest degree vector field in the sum, which is the linearization of the given vector field. Homogeneous stabilization of homogeneous systems is considered in [K4, K6, SA1], while connections between stabilizability and homogeneous feedback stabilization are explored in [H7, SA3].