We are interested in the asymptotic behaviour of the solutions of the system

\[

\begin{cases}

-\Delta u_i=f_i(x,u_i) - \beta u_i\sum_{j\neq i}a_{ij}u_j^2 & \text{in $\Omega$} \\

u_i>0 & \text{in $\Omega$}

\end{cases}

\qquad i=1,\dots,k

\]

in the limit of strong competition $\beta \to +\infty$. We prove that uniform $L^\infty$ boundedness of the solutions implies uniform boundedness of their Lipschitz norm. This extends known quasi-optimal regularity results and covers the optimal case for this class of problems. We present several applications of the uniform Lipschitz estimates, and finally we discuss extensions and several open problems in nonlocal context. The talk is based on joint works with Alessandro Zilio (EHESS).