Several new methods of in-beam conversion-electron and γ-ray spectrometry, applicable in the determination of E0 and E2 decay properties of low-lying 0+ states in even-mass nuclei, have been developed. The main attention has been paid to direct lifetime-measurement and coincidence methods based on the use of the natural pulsing of a cyclotron beam. With the aid of these methods, the similarity of the absolute decay rates of the two-neutron-hole 0+2 states in the N = 124 nuclei 206Pb and 208Po has been shown. A systematic investigation of the de-excitation of the 0+2 and 0+3 states in 112, 114, 116, 118, 120Sn has been carried out: twelve E0 transitions connecting the 0+ states have been observed, including very strong low-energy E0 transitions between the excited 0+ states, and several absolute transition probabilities have been determined. Furthermore, the new techniques have been applied successfully in determining the absolute E0 and E2 transition rates from the 0+2 and 0+3 states in 112Cd and 114Cd. The use of isotope-shift data in the calculation of the monopole strengths in 206Pb and 208Po is discussed. The results on even Sn and Cd nuclei are discussed within the framework of the coexistence of different shapes and of configuration mixing.