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Mergers and the spin of the dark matter halo are factors traditionallybelieved to determine the morphology of galaxies within a $\Lambda$CDMcosmology. We study this hypothesis by considering approximately 18,000 centralgalaxies at $z=0$ with stellar masses $M_{\ast} = 10^{9}-10^{12} \, {\rmM}_{\odot}$ selected from the Illustris cosmological hydrodynamic simulation.The fraction of accreted stars -- which measures the importance of massive,recent and dry mergers -- increases steeply with galaxy stellar mass, from lessthan 5 per cent in dwarfs to 80 per cent in the most massive objects, and theimpact of mergers on galaxy morphology increases accordingly. For galaxies with$M_{\ast} \gtrsim 10^{11} \, {\rm M}_{\odot}$, mergers have the expectedeffect: if gas-poor they promote the formation of spheroidal galaxies, whereasgas-rich mergers favour the formation and survivability of massive discs. Thistrend, however, breaks at lower masses. For objects with $M_{\ast} \lesssim10^{11} \, {\rm M}_{\odot}$, mergers do not seem to play any significant rolein determining the morphology, with accreted stellar fractions and mean mergergas fractions that are indistinguishable between spheroidal and disc-dominatedgalaxies. On the other hand, halo spin correlates with morphology primarily inthe least massive objects in the sample ($M_{\ast} \lesssim 10^{10} \, {\rmM}_{\odot}$), but only weakly for galaxies above that mass. Our results supporta scenario where (1) mergers play a dominant role in shaping the morphology ofmassive galaxies, (2) halo spin is important for the morphology of dwarfs, and(3) the morphology of medium-sized galaxies -- including the Milky Way -- showslittle dependence on galaxy assembly history or halo spin, at least when thesetwo factors are considered individually.

The role of mergers and halo spin in shaping galaxy morphology