We discuss possible conditions under which thermonuclear burning episodes in the hydrogen-rich envelopes of accreting white dwarfs give rise to outbursts similar in nature to those observed in the symbiotic stars AG Peg, RT Ser, RR Tel, AS 239, V1016 Cyg, V1329 Cyg, and HM Sge. In principle, thermonuclear runaways involving low-luminosity white dwarfs accreting matter at low rates produce configurations that evolve into A--F supergiants at maximum visual light and which resemble the outbursts of RR Tel, RT Ser, and AG peg. Very weak, nondegenerage hydrogen shell flashes on white dwarfs accreting matter at high rates (M> or approx. =10/sup -8/ M/sub sun/ yr/sup -1/) do not produce cool supergiants at maximum, and may explain the outbursts in V1016 Cyg, V1329 Cyg, and HM Sge. The low accretion rates demanded for systems developing strong hydrogen shell flashes on low-luminsoity white dwarfs are not compatible with observations of ''normal'' quiescent symbiotic stars. The extremely slow outbursts of symbiotic novae appear to be typical of accreting white dwarfs in wide binaries, which suggests that the outbursts of classical novae may be accelerated by the interaction of the expanding white dwarf envelope with its close binary companion.