The recoil, vacuum polarization and electron vertex corrections of first and second orders in the fine structure constant $alpha$ and the ratio of electron to muon and electron to alpha-particle masses are calculated in the hyperfine splitting of the
$1s^{(e)}_{1/2}2s^{(mu)}_{1/2}$ state of muonic helium atom (mu e ^4_2He) on the basis of a perturbation theory. We obtain total result for the muonically excited state hyperfine splitting $Delta u^{hfs}=4295.66$ MHz which improves previous calculations due to the account of new corrections and more accurate treatment of the electron vertex contribution.
We present new investigation of the Lamb shift (2P_{1/2}-2S_{1/2}) in muonic deuterium (mu d) atom using the three-dimensional quasipotential method in quantum electrodynamics. The vacuum polarization, nuclear structure and recoil effects are calcula
ted with the account of contributions of orders alpha^3, alpha^4, alpha^5 and alpha^6. The results are compared with earlier performed calculations. The obtained numerical value of the Lamb shift 202.4139 meV can be considered as a reliable estimate for the comparison with forthcoming experimental data.
