A group G is said to be automatic if the Cayley graph \(\Gamma\) with respect to some generating set X (where \(\Gamma\) is regarded as a metric space by giving each edge unit length) has the following property: there exists a constant \(k>0\) and a path in \(\Gamma\) for each \(g\in G\) (starting at 1 and ending at g) such that paths which end a distance 1 apart stay within a distance k of each other, and such that the words defined by the paths constitute a regular language in the free monoid on \(X\cup X^{-1}\). In their previous paper [Part I, ibid. 102, 305-334 (1990; Zbl 0714.20016)] the authors showed that the fundamental group of a piecewise Euclidean 2-complex of nonnegative curvature of type \(A_ 1\times A_ 1\) or \(A_ 2\) is automatic. \((A_ 1\times A_ 1\) corresponds to the Euclidean planar tesselation by unit squares, and \(A_ 2\) to the tesselation by equilateral triangles.) In the present paper the authors prove an analogous result for 2-complexes of types \(B_ 2\) and \(G_ 2\) corresponding to the Euclidean tesselations by (\(\pi\) /2,\(\pi\) /4,\(\pi\) /4) and (\(\pi\) /2,\(\pi\) /3,\(\pi\) /6) triangles respectively. They give some applications of their theorem to discrete groups of isometries of certain Euclidean buildings, and to connected sums of alternating links. In their previous paper the authors showed that a group given by a finite presentation satisfying the small cancellation conditions C(p), T(q) \(((p,q)=(6,3),(4,4),(3,6))\) is automatic, under the additional assumptions that all pieces have length 1 and no relator is a proper power. In the present paper the authors show that these additional assumptions can be dropped. Finally, the authors introduce the notion of a biautomatic group, and they show that such groups have solvable conjugacy problem.