Kernel-based approximation methods are important tools for data approximation on Euclidean domains and on spheres. Depending on whether one aims to study the theoretical properties of kernel methods or implement known methods efficiently, suitable representations of the kernels are needed. For example, a closed form representation is often suitable for efficient calculation, while a series or integral representation is used for theoretical investigation. We present new closed-form representations for compactly supported kernels in the Euclidean space and series representations for spherical kernels. In practice, RBFs defined in Euclidean space are frequently restricted to the sphere and used for the approximation of spherical data. Even though the strict positivity of the kernel on the sphere is directly inherited from the Euclidean kernel, error analysis does not follow immediately. To be able to employ most error estimates on the sphere, a representation of the kernel as a series expansion in spherical harmonics is required. We show two options for obtaining the required expansion coefficients and give examples of well-known basis functions.