Parametric filters, such as the Extended Kalman Filter and the Unscented Kalman Filter, typically scale well with the dimensionality of the problem, but they are known to fail if the posterior state distribution cannot be closely approximated by a density of the assumed parametric form. For nonparametric filters, such as the Particle Filter, the converse holds. Such methods are able to approximate any posterior, but the computational requirements scale exponentially with the number of dimensions of the state space. In this paper, we present the Coordinate Particle Filter which alleviates this problem. We propose to compute the particle weights recursively, dimension by dimension. This allows us to explore one dimension at a time, and resample after each dimension if necessary. Experimental results on simulated as well as real data confirm that the proposed method has a substantial performance advantage over the Particle Filter in high-dimensional systems where not all dimensions are highly correlated. We demonstrate the benefits of the proposed method for the problem of multi-object and robotic manipulator tracking.