This manuscript deals with the problem of observer design, to estimate all the states of a system. Let us consider the following multivariable system, \begin{align*} \dot{x}(t) & = A x(t)+B u(t),\\ y(t) & = C x(t),\\ z(t) & = L x(t), \end{align*} with the initial state \(x(0)=x_0\), \(x(t) \in \mathbb{R}^n\) is the state vector, \(u(t) \in \mathbb{R}^m\) is the known input, \(y(t) \in \mathbb{R}^p\) is the measurement output, and \(z(t) \in \mathbb{R}^r\) is the functional to be estimated. Matrix \(A \in \mathbb{R}^{n \times n}\) and matrix \(B \in \mathbb{R}^{n \times m}\). Matrices \(C \in \mathbb{R}^{p \times n}\) and \(L \in \mathbb{R}^{r \times n}\) are assumed to be of full row rank. The problem to be addressed in this paper is to find an observer for estimating the functional \(z(t)\). After the state of the art, the authors define functional observability and present the necessary and sufficient conditions for this type of observability. These conditions generalize those of observability and the Luenberger observer design in both full and reduced-order cases. The authors then establish the connection between functional observability and functional detectability (concept introduced by the authors [IEEE Trans. Autom. Control 68, No. 2, 975--990 (2023; Zbl 1541.93116)]), improving the frequency condition, which is necessary and sufficient for systems with semi-simple eigenvalues, by considering the more general case where the eigenvalues have different geometric and algebraic multiplicities. Section 3 of this paper focuses on the functional observer design, and in Section 4 the authors introduce a new method for designing an observer-based controller without the need to satisfy the observability condition. To demonstrate the effectiveness of the proposed scheme, they present two numerical examples.