The mathematical analyses of Cai et al. (1997) demonstrate the potential advantage of first order least-squares (FOLS) finite element method in providing a robust and efficient numerical platform for Stokes and elasticity problems. To evaluate the features of FOLS, I have taken the next step by implementing it and applying it to both incompressible viscous and inviscid transonic fluid flow problems. In the case of incompressible viscous flow, numerical solution suggests that one should use the velocity flux as independent variables to achieve optimal accuracy and efficiency. For transonic flows, FOLS has an implicit diffusion term that is proportional to the streamwise flow gradient and acts as a natural indicator for shocking capturing using local grid refinement solution strategy. Several pre-conditioned conjugate gradient solution procedures have been investigated for the resulting linear system; one that is matrix-free and another that requires the formation of element matrices. Both of them are perfectly suited for parallel computers. With the explicit formation of element matrix, one has more leverage to design pre-conditioners that can annihilate the adverse effect caused by large disparity in grid spacings and flow scales.