Massive multiple-input-multiple-output (MIMO) transmission is a promising technology to improve the capacity and reliability of wireless systems. However, the number of antennas that can be equipped at a base station (BS) is limited by the BS form factor, posing a challenge to the deployment of massive linear arrays. To cope with this limitation, this work discusses Full Dimension MIMO (FD-MIMO), which is currently an active area of research and standardization in the 3rd Generation Partnership Project (3GPP) for evolution towards fifth generation (5G) cellular systems. FD-MIMO utilizes an active antenna system (AAS) with a 2D planar array structure, which provides the ability of adaptive electronic beamforming in the 3D space. This paper presents the design of the AAS and the ongoing efforts in the 3GPP to develop the corresponding 3D channel model. Compact structure of large-scale antenna arrays drastically increases the spatial correlation in FD-MIMO systems. In order to account for its effects, the generalized spatial correlation functions for channels constituted by individual antenna elements and overall antenna ports in the AAS are derived. Exploiting the quasi-static channel covariance matrices of the users, the problem of determining the optimal downtilt weight vector for antenna ports, which maximizes the minimum signal-to-interference ratio of a multi-user multiple-input-single-output system, is formulated as a fractional optimization problem. A quasi-optimal solution is obtained through the application of semi-definite relaxation and Dinkelbach's method. Finally, the user-group specific elevation beamforming scenario is devised, which offers significant performance gains as confirmed through simulations. These results have direct application in the analysis of 5G FD-MIMO systems.