Some problems in the filtering and the detection of diffusion processes that are solutions of stochastic differential equations are studied. Transition densities for Markov process solutions of a large class of stochastic differential equations are shown to exist and to satisfy Kolmogorov's equations. These results extend previously known results by allowing the drift coefficient to be unbounded. With these results for transition densities the nonlinear filtering problem is discussed and the conditional probability of the state vector of the system conditioned on all the past observations is shown to exist and a stochastic equation is derived for the evolution in time of the conditional probability density. A stochastic differential equation is also obtained for the conditional moments. These derivations use directly the continuous time processes. Necessary conditions that coincide with the previously known sufficient conditions for the absolute continuity of measures corresponding to solutions of stochastic differential equations are obtained. Applications are made to the detection of one diffusion process in another. Previous results on the relation between detection and filtering problems are rigorously obtained and extended.