The brachistochrone problem is generalized for arbitrary velocity functions. The motion rather than the causes of the motion are considered with a kinematical analysis. In the most general form, the velocity is an arbitrary function of the independent and dependent variables as well as the slopes. Special cases of single variable velocity functions are then treated. For each case, some specific functions are considered as examples. The minimal paths are calculated using the principles of variational calculus. Curves are drawn which connects a given initial point to a final point. In some of the cases, the parametric solutions are given in the form of integrals and the constants appearing in the parametric solutions require a shooting like technique to allow the curves to reach their final destination points. The curves may be used in motion of land, marine and aerial vehicles if time is the most important factor in navigation.