The problem of steady, laminar, hydromagnetic simultaneous heat and mass transfer by natural convection flow over a vertical cone and a wedge embedded in a uniform porous medium is investigated. Two cases of thermal boundary conditions, namely the uniform wall temperature (UWT) and the uniform wall heat flux (UHF), are considered. A nonsimilarity transformation for each case is employed to transform the governing differential equations to a form whereby they produce their own initial conditions. The transformed equations for each case are solved numerically by an efficient implicit, iterative, finite-difference scheme. The obtained results are checked against previously published work on special cases of the problem and are found to be in excellent agreement. A parametric study illustrating the influence of the magnetic field, porous medium inertia effects; heat generation or absorption; lateral wall mass flux; concentration to thermal buoyancy ratio; and the Lewis number on the fluid velocity, temperature, and concentration as well as the Nusselt and the Sherwood numbers is conducted. The results of this parametric study are shown graphically and the physical aspects of the problem are discussed. It is concluded that while the local Nusselt number decreases owing to the imposition of the magnetic field, it increases as a result of the fluid's absorption effects. Also, both the local Nusselt and Sherwood numbers increase as the buoyancy ratio increases. This is true for both uniform wall temperature and heat flux thermal conditions. Furthermore, including the porous medium inertia effect in the mathematical model is predicted to decrease the local Nusselt number for both the isothermal and isoflux wall cases.