The dominion of a subalgebra H in an universal algebra A (in a class ) is the set of all elements such that for all homomorphisms if f, g coincide on H, then af = ag. We investigate the connection between dominions and quasivarieties. We show that if a class is closed under ultraproducts, then the dominion in is equal to the dominion in a quasivariety generated by . Also we find conditions when dominions in a universal algebra form a (...) lattice and study this lattice. (shrink)