Bridgeland stability conditions are powerful tools to study the derived categories of algebraic varieties. On smooth projective surfaces, the classical Bogomolov inequality gives a bound for the second Chern character of stable sheaves. This thus implies the existence of the Bridgeland stability conditions on surfaces by the tilting technique. However, to construct a Bridgeland stability condition on higher dimensional varieties, a Bogomolov-Gieseker type bound for higher Chern characters is expected, whose proof usually requires a stronger Bogomolov-Gieseker inequality. Besides Fano varieties and K3 surfaces, it is difficult to prove a stronger Bogomolov-Gieseker inequality. In this talk, I will start with definitions of Bridgeland stability conditions, and then show how to use the Bogomolov inequality to construct the Bridgeland stability conditions on surfaces. Then I will show one way to prove a stronger Bogomolov-Gieseker inequality on some Calabi-Yau threefolds, and as an application, the existence of the Bridgeland stability conditions on such Calabi-Yau threefolds. If time permits, I will mention another application of this stronger inequality by Feyzbakhsh-Thomas on enumerative invariants.