Antimagnetic rotation in weakly deformed nuclei Pd100,101, and chiral rotation in triaxial nuclei Nd135,136 have been investigated by means of gamma-ray spectroscopic measurements. These two modes of rotation are quite different from 'standard' rotational motion observed in well deformed nuclei. In these new modes, the total angular momentum either originates purely from a few particles in high-j orbitals, or these orbitals couple to collective motion of a triaxial core. Two cascades of four 'rotational-band-like' transitions were observed in Pd100 and Pd101, respectively, and have been proposed as corresponding to antimagnetic rotation, based on the observed spectroscopic properties and a comparison with calculations in the configuration-dependent cranking Nilsson-Strutinsky formalism. Two Delta(I)=1 bands with close excitation energies and the same parity were observed in Nd135. These bands are directly linked by Delta(I)=1 and Delta(I)=2 transitions. The chiral nature of these two bands is confirmed by comparison with three-dimensional tilted axis cranking calculations. This is the first observation of a three-quasiparticle chiral structure and establishes the primarily geometric nature of this phenomenon. Although there are no chiral partner bands observed in even-even nucleus Nd135 from our data, TAC calculations indeed give chiral solutions related to several Delta(I)=1 bands in the nucleus Nd136.