The paper is devoted to the study of a class of continuous-time uncertain (CTU) systems which satisfy a certain Integral Quadratic Constraint (IQC). The problems of robust filtering, robust prediction, and robust smoothing for CTU systems satisfying the IQC are defined. Section 3 of the paper gives the solution to the problem of robust filtering in terms of a differential matrix Riccati equation, and the set of possible states is shown to be an ellipsoid. The problem of robust filtering for time-invariant uncertain systems is also addressed. Section 4 deals with the problem of robust smoothing, divided into the following three subproblems: (RS1) robust fixed interval smoothing, (RS2) robust fixed lag smoothing, and (RS3) robust fixed point smoothing. Section 5 focuses on the (RS1) problem, while Section 6 addresses the closely related problem of robust observability of uncertain systems. Necessary and sufficient conditions for robust observability are given in terms of existence of a solution to a differential matrix Riccati equation. The final Section 7 illustrates by examples the proposed analysis.