Hamilton equations of motion for mechanical systems are well-known from theoretical mechanics. In the classical approach they are defined for the first order regular variational problem by the regularity of the matrix formed by second order partial derivatives of the Lagrange function with respect to m velocities (for a mechanical system with m degrees of freedom). In such a case we obtain 2m first order ODE of motion (Hamilton equations) instead of m second order Euler-Lagrange equations. Compared to standard approaches in Hamilton theory, our approach presented in this chapter is generalized by the new definition of regularity and its consequences, including examples. The standard definition of regularity based on the properties of the Lagrangian is generalized by introducing the so-called Hamilton extremals and so called mechanical system as the mathematical concept. As a result, regularity is defined on the base of the Euler-Lagrange form instead of the Lagrangian (various Lagrangians lead to the same Euler-Lagrange form.) The basic idea of the Hamilton-Jacobi theory is presented.