  Regular chains and triangular decompositions are fundamental and
well-developed tools for describing the complex solutions of polynomial
systems. This paper proposes adaptations of these tools focusing on solutions
of the real analogue: semi-algebraic systems. We show that any such system can
be decomposed into finitely many {\em regular semi-algebraic systems}. We
propose two specifications of such a decomposition and present corresponding
algorithms. Under some assumptions, one type of decomposition can be computed
in singly exponential time w.r.t.\ the number of variables. We implement our
algorithms and the experimental results illustrate their effectiveness.
