Let $V(d,m,k)$ be the variety of plane projective irreducible curves of degree $d$ with $m$ nodes and $k$ cusps as their only singularities. We prove that $V(d,m,k)$ is non-empty, non-singular and irreducible when $m+2k< \alpha d^2$, where $\alpha $ is some absolute explicit constant. This estimate is optimal with respect to the exponent of $d$