The scalar field models with the Lagrangian L = F(X) − V(), which we call general non-canonical scalar field models, are investigated. We find that a special square potential (with a negative minimum) is needed to drive the linear field solution ( = ₀t) in our model, while in the K-essence model (L = −V()F(X)) the potential should be taken as an inverse square one. Hence the cosmological evolutions for these models are totally different. The linear field solutions are found to be highly degenerate, and their cosmological evolutions are equivalent to the model where the sound speed diverges. We also study the stability of the linear field solution and find the condition for stable solutions to exist. The cosmological solution in the presence of matter and radiation is further studied by numerically solving the potential and the cosmological evolution, and the results are shown to be quite different from the case of no matter or radiation. Then we analyze the case with a constant barotropic index γ and show that, unlike in the K-essence model, the detailed form of F(X) depends on the potential V(), and that this constant γ solution is stable for γ₀ ≤ 1. When the potential is taken to be a constant, we find the first integral and obtain the corresponding γ, which is similar to that in the K-essence model.