We suggest three superpositions of COGARCH (sup-CO-GARCH) volatility processes driven by Lévy processes or Lévy bases. We investigate second-order properties, jump behaviour, and prove that they exhibit Pareto-like tails. Corresponding price processes are defined and studied. We find that the sup-CO-GARCH models allow for more flexible autocovariance structures than the COGARCH. Moreover, in contrast to most financial volatility models, the sup-CO-GARCH processes do not exhibit a deterministic relationship between price and volatility jumps. Furthermore, in one sup-CO-GARCH model not all volatility jumps entail a price jump, while in another sup-CO-GARCH model not all price jumps necessarily lead to volatility jumps.