In the early days of computing, geophysical fluid dynamics (GFD), numerical weather prediction (NWP) in particular, was a dominant factor in the design of computer architecture and algorithms. This early work focussed initially on finite difference algorithms on a rectangular computational grid and later on spectral methods. After the initial work of Charney, von Neumann, and Arakawa however, the focus shifted from the basic algorithms for the numerical solution of the fundamental differential equations to improvements in the model physics. Further work on fundamental numerical algorithms shifted to other disciplines - predominately the then emerging aerospace community. As a result, for 40 years, the GFD community has been using numerical techniques that are virtually unchanged. The two primary numerical methodologies that have been used for modeling the atmosphere and the ocean have been spectral methods for global modeling and structured rectilinear grids for regional modeling. In the last few years, a new paradigm for atmospheric and oceanic simulation has emerged: adaptive, unstructured, triangular grids. This paradigm has the advantage of tremendous flexibility in providing high resolution where required by either static physical properties (terrain elevation, coastlines, land use) or the evolving dynamical situation. The first application of this paradigm was the Operational Multiscale Environment model with Grid Adaptivity (OMEGA), an atmospheric simulation and forecasting tool; a more recent application of this paradigm is the more recently developed Multiscale Ocean Simulation System (MOSS). OMEGA with its embedded Atmospheric Dispersion Model (ADM) is a new atmospheric simulation system for real-time hazard prediction, conceived out of a need to advance the state-of-the-art in numerical weather prediction in order to improve our capability to predict the transport and diffusion of hazardous releases. MOSS is a nascent implementation of a similar grid structure and numerical algorithm to the simulation of ocean circulation. The purpose of this paper is to provide a description of this new paradigm and to present its use in atmospheric and oceanic simulation.