The Madden–Julian Oscillation (MJO), a planetary-scale disturbance in zonal winds and equatorial convection that dominates intraseasonal variability in the Tropics, is a challenge to explain and notoriously difficult to simulate with conventional climate models. This study discusses numerical experiments conducted with a novel Lagrangian atmospheric model (LAM) that produce surprisingly robust and realistic MJOs, even at very low resolution. The LAM represents an atmosphere as a collection of conforming air parcels with motions that are predicted using Newtonian mechanics. The model employs a unique convective parametrization, referred to as Lagrangian overturning (LO), in which air parcels exchange vertical positions in convectively unstable regions. A key model parameter for simulating MJOs is the mixing between adjacent ascending and descending parcels, with more frequent and stronger MJOs occurring when greater mixing is prescribed. Sensitivity tests suggest that MJOs simulated with the LAM are not particularly sensitive to model resolution, but their structure and propagation speed do depend on sea-surface temperatures, large-scale precipitation patterns and surface fluxes. An important conclusion of this article is that the most fundamental dynamics of the MJO are captured by the LO convective parametrization coupled with large-scale atmospheric circulations.