Reset control systems are a special type of impulsive hybrid systems in which the time evolution depends both on continuous dynamics between resets and the discrete dynamics corresponding to the reset instants. In this work, stability of reset control systems is approached by using an equivalent (time-varying) discrete time system, introducing new sufficient stability conditions that explicitly depend on the reset times. The stability conditions overcome previous results such as the H-beta-condition and can be also applied to reset systems with unstable base system. In addition, stabilization of reset control systems is approached for systems with stable base system, resulting in that a reset control system can be always stabilized if reset intervals are forced to be large enough, avoiding resets at some crossings if necessary.