This thesis consists of six chapters. The first one is devoted to preliminaries. In Chapter 2, the author proves the solvability of variational inequalities of the form \[ \langle Au +Gu-f,T_k(v-u)\rangle \geq 0 \quad \forall v\in K^p\cap L^\infty(\Omega), \] where \(A\) is a Leray–Lions operator defined in \(W^{1,p}_0(\Omega)\), \(\langle Au,v\rangle=\int_\Omega a(x,u,\nabla u)\nabla v\,dx\), \(Gu=g(x,u,\nabla u)\), \(f\in L^1(\Omega)\), \(T_k(s):=\max(-k,\min(k,s))\), and \(K^p:=\{v\in W^{1,p}_0(\Omega):v\geq\psi\text{ a.e.~in }\Omega\}\). Here \(\Omega\subset\mathbb R^N\) is bounded, \(2-1/N<p\leq N\), \(g(x,s,\zeta)s\geq0\) and \(| g(x,s,\zeta)| \leq b(| s| )(c(x)+| \zeta| ^p)\), \(c\in L^1(\Omega)\), and \(\psi^+\in W^{1,p}_0(\Omega)\cap L^\infty(\Omega)\). The solution \(u\) is required to belong to \(K^q\) for any \(q<N(p-1)/(N-1)\) and \(T_k(u)\in W^{1,p}_0(\Omega)\) for any \(k>0\). The proof is based on an approximation of \(f\) by regular functions and a priori estimates of approximating solutions. A similar problem is studied in Chapter 3 in the framework of Orlicz–Sobolev spaces. In Chapter 4, a penalisation method is used in order to solve a variational inequality of the form \[ u\in K^p:\qquad \langle Au-f,v-u\rangle \geq 0 \quad \forall v\in K^p, \] where \(A\) is as above, \(f\in W^{-1,p'}(\Omega)\), \(K^p=\{v\in W^{1,p}_0(\Omega):q_-\leq v\leq q_+\text{ a.e.}\}\) and \(q_\pm\) are of the form \(q_\pm(x)=\pm\inf\{s>0:\pm g(x,\pm s)\geq1\}\) for suitable \(g\). Chapter 5 is devoted to the same problem with \(f\in L^1(\Omega)\). In this case, the test function \(v-u\) is replaced with the truncation \(T_k(u-v)\) and the solution \(u\) is required to belong to \(K^q\) for any \(q<N(p-1)/(N-1)\), \(T_k(u)\in W^{1,p}_0(\Omega)\) for any \(k>0\). In Chapter 6, the author studies a parabolic analogue of the above problem.