The core of this paper deals with the construction of input-decoupled observers which seek asymptotic estimation of a desired output variable (a linear combination of state and input) of a time-invariant either continuous- or discrete-time system driven by unknown inputs and disturbances. Exact, almost, optimal (suboptimal) and constrained optimal estimation or filtering problems are formulated and studied. All the problems defined and studied here are inherently interconnected, and have a strong common thread of estimation and filtering in the face of the unknown input and external disturbance signals. They are interconnected from a variety of angles, e.g. they are motivated by one another, methods of obtaining the solvability conditions and their methods of solution rely on one another, etc. Thus, a hierarchy of problems and their solutions is built on top of one another. Some of the problems studied here are known in the literature but not in as general a form as is given here, while a majority of the problems studied here are new. A classical variation of all the above problems is also studied here by introducing an e-step delay in estimating the desired output from the measured output. The underlying philosophy throughout this work has been to study most if not all of the facets of estimation and filtering in one stretch under a single folder. Our study of all the above problems has been guided by three important perspectives: (1) obtaining the solvability conditions, both necessary and sufficient; (2) obtaining optimal performance whenever it applies; and (3) developing sound methodologies to design and construct appropriate observers or filters.