This paper introduces in detail a new systematic method to construct approximate finite-dimensional solutions for the nonlinear filtering problem. Once a finite-dimensional family is selected, the nonlinear filtering equation is projected in Fisher metric on the corresponding manifold of densities, yielding the projection filter for the chosen family. The general definition of the projection filter is given, and its structure is explored in detail for exponential families. Particular exponential families which optimize the correction step in the case of discrete-time observations are given, and an a posteriori estimate of the local error resulting from the projection is defined. Simulation results comparing the projection filter and the optimal filter for the cubic sensor problem are presented. The classical concept of assumed density filter (ADF) is compared with the projection filter. It is shown that the concept of ADF is inconsistent in the sense that the resulting filters depend on the choice of a stochastic calculus, i.e. the Itô or the Stratonovich calculus. It is shown that in the context of exponential families, the projection filter coincides with the Stratonovich-based ADF. An example is provided, which shows that this does not hold in general, for non-exponential families of densities.