In the presence of P-violating interactions, the exchange of vector bosons between electrons and nucleons induces parity-nonconserving (PNC) effects in atoms and molecules, while the exchange of vector bosons between nucleons induces anapole moments of nuclei. We perform calculations of such vector-mediated PNC effects in Cs, Ba + , Yb, Tl, Fr and Ra + using the same relativistic many-body approaches as in earlier calculations of standard-model PNC effects, but with the longrange operator of the weak interaction. We calculate nuclear anapole moments due to vector boson exchange using a simple nuclear model. From measured and predicted (within the standard model) values for the PNC amplitudes in Cs, Yb and Tl, as well as the nuclear anapole moment of 133 Cs, we constrain the P-violating vector-pseudovector nucleon-electron and nucleon-proton interactions mediated by a generic vector boson of arbitrary mass. Our limits improve on existing bounds from other experiments by many orders of magnitude over a very large range of vector-boson masses.PACS numbers: 31.30.jg,11.30.Er,32.60.+i,11.40.Ha Introduction. -The discovery that the parity symmetry (that is, the symmetry associated with the inversion of the spatial coordinates) is not conserved by the weak interaction [1, 2] was pivotal for the subsequent development of the standard model (SM) of particle physics, which to date remains the most successful description of elementary particles and their interactions. At the same time, the SM does not explain a number of important observed phenomena, such as dark matter, suggesting the existence of at least one new feeblyinteracting particle beyond the SM.Atomic parity nonconservation (PNC) experiments provide a very powerful and relatively inexpensive test of the SM at low energies [3][4][5]. Measurements and calculations (within the SM) of the Cs 6s-7s PNC amplitude have to date provided the most precise atomic test of the electroweak theory [6][7][8][9][10][11][12][13][14][15][16] and also invaluable information on parity-violating interactions within the nucleus [10,[17][18][19][20]. Investigations of atomic PNC phenomena have been applied to search for new vector bosons with masses greater than 100 keV [21][22][23][24][25], as well as interactions of electrons and nucleons with bosonic dark matter and dark-energy-type fields [26][27][28].In the present work, we investigate the manifestation of the exchange of a generic vector boson Z of arbitrary mass between atomic electrons and nucleons, in the presence of the following couplings [29]: