This paper is concerned with the following initial value problem (IVP) for the 1-D semilinear Schrödinger equation: \[ i\partial_tu= i\partial^2_xu+N_j(u,\overline u),\quad x,t\in\mathbb{R},\quad u(x,0)=u_0(x), \] where \(N_1(u,\overline u)= c_1uu\), \(N_2(u,\overline u)=c_2u\overline u\), \(N_3(u,\overline u)=c_3\overline u\overline u\). The authors study local well posedness in the classical Sobolev space \(H^s\) of the associated IVP and the periodic boundary value problem (PBVP). Their main interest is to obtain the lowest value of \(s\) which guarantees the desired local well posedness result. They prove that at least for the above quadratic cases these values are negative and depend on the structure of nonlinearity considered. For \(s,b\in\mathbb{R}\), let \(X_{s,b}\) denote the completion of the Schwartz class \(S(\mathbb{R}^2)\) with respect to the norm \[ |F|_{X_{s,b}}= \Biggl(\int^\infty_{-\infty}\int^\infty_{-\infty}(1+|\tau-\xi^2|)^{2b}(1 +|\xi|)^{2s}|\widehat F(\xi,\tau)|^2d\xi d\tau\Biggr)^{1/2}. \] The authors prove the local well posedness for (IVP) in each case. The main part of the results is the following: In the case \(N_1(u,\overline u)\): For \(s\in(-3/4,0]\), there exists \(b\in(1/2,1)\) such that for any \(u_0\in H^s(\mathbb{R})\) the local well posedness is proved for a solution \(u\in C([-T,T], H^s(\mathbb{R})\cap X_{s,b})\). Analogously, in the cases \(N_2(u,\overline u): s\in(-1/4,0]\), \(N_3(u,\overline u): s\in(-3/4,0]\). They also obtain the well posedness of (PBVP) for the equation given above. The authors give a clear survey of results on these fields, especially for cases where the nonlinear term is \(N(z_1,z_2)= \sum_{|\alpha|\leq 5}a_\alpha z^{\alpha_1}z^{\alpha_2}_2\).