We study the spreading of spheroidal aggregates of cells, expressing a tunable level of E-cadherin molecules, on glass substrates decorated with mixed fibronectin and polyethylene glycol. We observe the contact area by optical interferometry and the profile by side-view microscopy. We find a universal law of aggregate spreading at short times, which we interpret through an analogy with the spreading of viscoelastic droplets. At long times, we observe either partial wetting or complete wetting, with a precursor film of cells spreading around the aggregate with two possible states. In strongly cohesive aggregates this film is a cellular monolayer in the liquid state, whereas in weakly cohesive aggregates, cells escape from the aggregate, forming a 2D gas. The escape of isolated cells is a physical mechanism that appears also to be present in the progression of a noninvasive tumor into a metastatic malignant carcinoma, known as the epithelial-mesenchymal transition.