In a pioneering paper, on 1910, S. Zaremba established the \textit{strong maximum principle} saying that if $\Omega$ is a smooth bounded domain in $% \mathbb{R}^{N}$ and a function $u$ verifies \begin{equation}\left\{\begin{array}{lr} -\Delta u\geq f(x) & \text{in }\Omega , \\u=0 & \text{on }\partial \Omega ,%\end{array}%\right. \label{ProbLineal}\end{equation}% \noindent then \begin{equation} u(x)>0\text{ in }\Omega , \tag{P$_{u}$} \label{Positiv u} \end{equation} \noindent assumed that \begin{equation} f(x)\geq 0\text{ in }\Omega ,\text{ }f\neq 0. \tag{P$_{f}$} \label{Positiv f} \end{equation} The extension to a more general second order elliptic operator was due to E. Hopf, on 1927. Moreover, some years later, on 1952, E. Hopf and O.A. Oleinik, independently, proved that under the above conditions the normal derivative of $u$ satisfies the following sign condition \begin{equation} \frac{\partial u}{\partial n}<0\text{ on }\partial \Omega . \label{Hypo2.3} \end{equation}. The main goal of this lecture is to show that the non-changing sign assumption (\ref{Positiv f}) can be removed in a suitably way (for instance when $f(x)<0$ in some neighborhood of $\partial \Omega $)$.$ Under such type of new assumptions on $f$ we can prove that: (A) the positivity of $u,$ property (\ref{Positiv u}), still holds and $% \frac{\partial u}{\partial n}\leq 0$ on $\partial \Omega ,$ \noindent and (B) under additional conditions on $f(x)$, the positive \textit{solution} of the equation (\ref{ProbLineal}) [i.e., now with the identity symbol $=,$ instead $\geq $] does not satisfy (\ref{Hypo2.3}) but $\frac{\partial u}{% \partial n}=0$ on $\partial \Omega $ \ (this corresponds to the notion of \textit{flat solutions} already considered by different authors in the framework of some nonlinear problems). The above extension of the strong maximum principle admits many generalizations (nonlinear elliptic operators, zero and first order terms, etc.). In the case of linear parabolic problems%\[ \left\{\begin{array}{lr} u_{t}-\Delta u=f(x,t) & \text{in }\Omega \times (0,+\infty ), \\ u=0 & \text{on }\partial \Omega \times (0,+\infty ), \\ u(x,0)=u_{0}(x) & \text{on }\Omega ,% \end{array}% \right.\]% it can be proved that, under suitably changing sign conditions on $u_{0}(x)$ and/or on $f(x,t)$ (even if it occurs for any $t>0$), we get the global positivity of $u(x,t)$ on $\Omega $, for large values of time ( $\exists $ $% t_{0}\geq 0$ such that $u(x,t)>0$ a.e. $x\in \Omega ,$ for any $t>t_{0}$). An application to some \textit{sublinear} \textit{indefinite} semilinear equations will be also given. [Joint results with Jes\'{u}s Hern\'{a}ndez].