Technical details of a new global mapping technique for finding equilibrium (EQ) and transition structures (TS) on potential energy surfaces (PES), the scaled hypersphere search (SHS) method (Ohno, K.; Maeda, S. Chem. Phys. Lett. 2004, 384, 277), are presented. On the basis of a simple principle that reaction pathways are found as anharmonic downward distortions of PES around an EQ point, the reaction pathways can be obtained as energy minima on the scaled hypersphere surface, which would have a constant energy when the potentials are harmonic. Connections of SHS paths between each EQ are very similar to corresponding intrinsic reaction coordinate (IRC) connections. The energy maximum along the SHS path reaches a region in close proximity to the TS of the reaction pathway, and the subsequent geometry optimization from the SHS maximum structure easily converges to the TS. The SHS method, using the one-after-another algorithm connecting EQ and TS, considerably reduces the multidimensional space to be searched to certain limited regions around the pathways connecting each EQ with the neighboring TS. Applications of the SHS method have been made to ab initio surfaces of formaldehyde and propyne molecules to obtain systematically five EQ and nine TS for formaldehyde and seven EQ and 32 TS for propyne.