Almost every type of investment comes with risk. The level of risk you assume with a mutual fund depends on the investments you select (based on your investment objective, risk tolerance, need for liquidity, and market conditions).
Variable annuity risk is similar-it depends on the investments you select in the annuity subaccount, and whether your contract comes with a guaranteed death benefit or other distribution features.
With fixed annuities, there are different risks. The insurance company issuing the fixed annuity will likely guarantee that your money earns a certain rate of interest for a specific period. This guaranteed rate may be adjusted up or down by the insurance company as you enter each new guarantee period. However, the contract will contain a minimum interest rate guarantee.