In this project we study gravity in 2+1 dimensions. With a fewer number of degrees of freedom, 2+1 dimensional problem is less complicated than the 3+1 dimensional one. We study Anti-de sitter space, which is a solution to the Einstein's field equation with negative cosmological constant. It is known that BTZ black hole can be obtained by identifying one of the direction anti-de Sitter space. The 2+1 dimensional black hole is very much alike to what Kerr black hole is; it has an inner and an outer horizon and it is described by a specified mass and angular momentum. Hence the gravity in 2+1 dimensions can be considered as a useful model for better understanding of the 3+1 dimensional gravity. We review the geometry of wormholes, Lorentzian and Euclidian, as another justify; TEXT-INDENT: 36pt; MARGIN: 0cm 0cm 10pt; unicode-bidi: embed; DIRECTION: ltr" In the BTZ black hole, one encounters closed time-like curves in casual structure space-time. One way to resolve this problem is to introduce O-BTZ solution. The structure of the O-BTZ is a orbifolding of the space-time. In this geometry there is no closed time-like curves; therefore, the structure of O-BTZ is a causally complete. We investigate the scalar field with conformal coupling in the BTZ and O-BTZ background. Using the two point function and by applying the projection method, we achieve the two point function of BTZ space-time. Using this two point function we find the vacuum expectation value of the energy-momentum tensor. Our motivation in calculating this quantity in the rotating black hole backgrounds is to study the quantum stability of black hole horizons. We see that this quantity at the inner horizon of the BTZ diverges. This shows that the BTZ inner horizon is not stable in the presence of vacuum quantum fluctuations of the scalar field. Whilst by calculating the vacuum expectation value in the O-BTZ background, this quantity is finite everywhere. This shows that the probe approximation is well defined in the O-BTZ background.