The effect of elasticity on the meniscus shape and film thickness for the free boundary creeping flow created by injecting air to a Hele-Shaw cell initially filled with a viscoelastic fluid is studied theoretically. The theory is developed with the assumptions that the displaced viscoelastic fluid wets the walls and that capillary number Ca and the local Weissenberg number We are both small. The transition region between the advancing meniscus and the entrained film is where the fluid theology has its greatest effect. The Oldroyd-B constitutive equation is used to model the viscoelastic fluid. The theory is formulated as a double expansion in Ca-1/3 and We/Ca-1/3. According to our asymptotic analysis, as the fluid becomes more viscoelastic, the film thickness decreases and the pressure drop at the meniscus tip increases. A detailed analysis shows that the dominant mechanisms are the resistance to stream-wise strain, tending to lower the film thickness, and the buildup of shear stress, tending to raise the film thickness, with the former being the numerically larger of the two. Our theory leads to the prediction that as viscoelasticity comes to dominate the shear resistance, the film thickness will scale with U-4/3. The effects of shear thinning and normal stress thinning are analyzed by adapting an approximate model obtained by retaining only the dominant terms in the force balance. Our Hele-Shaw cell theory is extended to dip coating and soap film forming flows. Our theoretical results pertaining to the effects of viscoelasticity on the thickness of the film are in qualitative agreement with recent experimental data.