How to implement seismic or any other type of ray-based tomography in a new class of cellular parameterization built from any chosen distribution of nodes in two or three dimensions is shown. The complete flexibility of this novel approach allows one to introduce detail in a tomographic model only where desired, thereby reducing memory and computation time, or to impose complex a priori constraints on the inversion. Full details of powerful new algorithms are given to generate unequally sized tetrahedral or polyhedral cells and to calculate the necessary Frechet derivatives required in linearized tomography. These algorithms are efficient enough in three dimensions, to allow the parameterization to be refined during an inversion. The methods are illustrated with numerical examples. It is concluded that in linear or nonlinear inversion the computational cost of the new algorithms will not be significantly higher than that incurred by using a regular Cartesian grid. ¿ 1998 American Geophysical Union