This　paper　introduces　a　fractal-fractional　version　of　the　nonlinear　2D　advection-diffusion　equation　and　proposes　a　meshless　method　based　on　the　moving　least　squares　shape　functions　for　its　numerical　solution.　The　fractal-fractional　derivative　in　the　Atangana-Riemann-Liouville　is　considered　to　define　this　equation.　The　proposed　method　includes　the　following　steps:　We　first　approximate　the　fractal-fractional　derivative　using　the　finite　differences　method　and　derive　a　recursive　algorithm　by　applying　the　θ-weighted　method.　Next,　using　the　moving　least　squares　shape　functions,　we　expand　the　solution　of　the　problem　and　its　corresponding　partial　derivatives　and　substitute　them　into　the　recurrence　formula.　Finally,　in　accordance　with　the　previous　step,　we　obtain　a　linear　system　of　algebraic　equations　which　must　be　solved　at　each　time　step.　The　validity　and　accuracy　of　the　method　are　investigated　by　solving　some　numerical　examples.　©　2021