This paper proposes an efficient algorithm for the enumeration of every regular triangulation of a given point set $V = \{v_1,...,v_n\} \subset \boldmath{R^{d-1}}$.Regular triangulations from a subclass of triangulations and can be defined as a natural extension of the Delaunay triangularion, investigated frequently in computational geometry, and of lexicographic trianglations,a subclass of triangulations well-known in the theory of oriented matroids. Moreover, regular triangulations have interesting algebraic aspects in connection with a famous paradigm of computer algebra,Gr\"obner bases. It must be mentioned that the enumeration of regular triangulations of products of simplices is also an important problem in mathematics. Our algorithm achieves a reduced space complexity $O(ds)$, where $s$ is the upper bound of the number of simplices contained in one regular triangulation, i.e., $O(n^{\lfloor{\frac{d}{2}}\rfloor})$. The space complexity of the algorithm reported previously by Billera,et al. was $\Theta(n^{(d+1)(r-1)})$, although their algorithm completes the enumeration in the worst case optimal time, $\Theta(n^{(d+1)(r-1)})$, where $r$ is defined to be $n-d$. Therefore, the space complexity of our algorithm can be taken as a drastic improvement. The time complexity of our algorithm is $O(r^3s^2l(s,r)T)$,where $l(s,r)$ denotes the time required for solving a linear programming problem with $s$ constraints in $r$ variables, and $T$ is the number of regular triangulations of $V$,which is bounded from above by $O(n^{(d+1)(r-1)})$.Our time complexity is proportional to the output size $T$. Since it is shown that the upper bound of $T$ is attained by quite special point configurations,the output size sensitivity may work positively in most cases.