The main objective of this paper involving neutral stochastic functional differential equations (NSFDEs), with finite or infinite history dependence, is to prove the existence and uniqueness of their global solutions by imposing conditions that hold for highly non-linear NSFDEs' coefficients. For that purpose, Yamada-Watanabe condition or the local Lipschitz condition for the drift and diffusion coefficients are imposed, together with contractivity condition for the neutral term. Also, instead of the linear growth condition for the drift and diffusion coefficients of the equations, generalized Khasminskii-type conditions are applied. The proof of the existence and uniqueness of the solution also leads us to estimates of the moments to the solution. Consequently, we discuss some asymptotic properties of the solution in terms of the generalized Lyapunov exponent. Additionally, we consider a class of neutral stochastic differential equations with state-dependent delay, as a special case of NSFDEs. The theoretical results are illustrated with two examples.