Greg Chini University of New Hampshire Buoyancy-driven convection in fluid-saturated porous media is a key environmental and technological process, with applications ranging from CO2 storage in terrestrial aquifers to the design of compact heat exchangers. Porous medium convection is also a paradigm for forced--dissipative infinite-dimensional dynamical systems, exhibiting spatiotemporally chaotic dynamics if not “true’’ turbulence. In this talk, I will summarize our investigations of the dynamical structure and heat transport in porous medium convection between isothermal plane parallel boundaries at asymptotically large values of the Rayleigh number Ra. These studies employ a complement of direct numerical simulations (DNS) secondary stability theory, and variational analysis. Our DNS confirm the remarkable tendency for the interior flow to self-organize into closely spaced columnar plumes that are well described by a small number of (horizontal) Fourier modes. More complex spatiotemporal features are confined to -- and the convective transport ultimately determined by the dynamics of -- thermal boundary layers (BLs) near the heated and cooled walls. The relatively simple form of the interior flow motivates investigation of unstable steady convective states at large Ra as a function of the domain aspect ratio L. Analysis of the secondary instabilities of these equilibrium states sheds light on nonlinear scale selection in “turbulent’’ porous medium convection. Finally, an adaptation of a variational methodology for obtaining rigorous bounds on heat transport in convective flows is shown to yield quantitatively useful predictions of the Nusselt number Nu and to furnish a functional basis that is naturally adapted to the BL dynamics at large Ra.