This paper presents the multivariable extension of the feedback particle filter (FPF) algorithm for the nonlinear filtering problem in continuous-time. The FPF is a control-oriented approach to particle filtering. The approach does not require importance sampling or resampling and offers significant variance improvements; in particular, the algorithm can be applied to systems that are not stable. This paper describes new representations and algorithms for the FPF in the general multivariable nonlinear non Gaussian setting. Theory surrounding the FPF is improved: Exactness of the FPF is established in the general setting, as well as well-posedness of the associated boundary value problem to obtain the filter gain. A Galerkin finite-element algorithm is proposed for approximation of the gain. Its performance is illustrated in numerical experiments. (C) 2016 Elsevier Ltd. All rights reserved.