  A vertex $v$ in a map $M$ has the face-sequence $(p_1 ^{n_1}. \ldots.
p_k^{n_k})$, if there are $n_i$ numbers of $p_i$-gons incident at $v$ in the
given cyclic order, for $1 \leq i \leq k$. A map $M$ is called a semi-equivelar
map if each of its vertex has same face-sequence. Doubly semi-equivelar maps
are a generalization of semi-equivelar maps which have precisely 2 distinct
face-sequences. In this article, we enumerate the types of doubly
semi-equivelar maps on the plane and torus which have combinatorial curvature
0. Further, we present classification of doubly semi-equivelar maps on the
torus and illustrate this classification for those doubly semi-equivelar maps
which comprise of face-sequence pairs $\{(3^6), (3^3.4^2)\}$ and $\{(3^3.4^2),
(4^4)\}$.
