Here are their pros and cons vis-a-vis bank deposits
Investing in Debt Mutual Funds are equally or more safe than any individual Fixed Deposits; but its daily NAV and interest sensitive price movement makes it fluctuate in its market value, sometimes. However if you are looking to invest in 3 or more years, such schemes offer better propositions .
Because of its Open Ended nature, units of such schemes can be redeemed anytime at the prevailing NAV. Generally such Debt funds have some exit load or charges if redeemed before 1 year or so. One should check exit charges before investing.
It is said in every MF investment disclaimer, that past returns are not the guarantee for future performance. Still, historically if you check the returns of these schemes, you will find a performance better than traditional Fixed Deposits.