Convection is essential and ubiquitous in Nature. For a century, classical thermal convection, or Rayleigh-Bénard convection, has been a central paradigm for laboratory studies. This thesis concerns an electrical analogue of Rayleigh-Bénard convection — electroconvection in a thin fluid film. This complementary system was studied with a combination of experiment, theory, and numerical simulation. The fluid film is driven to convect by a critical applied electric potential interacting with a charge inversion, in direct analogy with the buoyancy inversion that drives thermal convection. As the imposed voltage is increased, electroconvection proceeds from steady, laminar patterns through time-dependent flows and eventually into chaotic and turbulent regimes. The experimental procedure consisted of precise measurements of current-voltage (IV) characteristics when a DC voltage was applied to an annular film between two concentric electrodes. The onset of convection was found by a change in the slope of the IV curve;​ unsteady flow was indicated by a large increase in current fluctuations. From the IV measurements, the corresponding dimensionless charge transport, or Nusselt number Nu, was determined as a function of the electric forcing, characterized by the dimensionless Rayleigh number ℛ. A power-law relationship Nu ~ℛ^γ was observed in the turbulent convection regime when ℛ ≳ 10⁴. The influence of the annular geometry, characterized by aspect ratio Γ, on this scaling was investigated. A scaling theory was developed which explains the power law and its dependence on Γ. The corresponding theory for thermal convection does not account for this Γ dependence. In addition, a direct numerical simulation was constructed using a pseudo-spectral method, based on realistic governing equations. The simulation affords deep insights into the flow dynamics, charge distribution and electric potential of the electroconvection instability and its route to turbulence, including for the case of an externally applied shear.