The main results of the paper are concerned with general modular approximation theorems in modular spaces for a net of operators \(\{T_w\}\) of type \[ (T_wf)(s)=\int _GK_w(s,t,f(t)) d\mu (t),\quad s\in G, \] where \(G\) is a locally compact topological group, \(\mu \) is the Haar measure on the Borel \(\sigma \)-algebra \({\mathcal B}\) of \(G\), and where \(w\in {\mathcal W}\), \({\mathcal W}\) a set of indices, \(K_w\) is a kernel satisfying a Lipschitz condition of type \[ |K_w(s,t,u)-K_w(s,t,v)|\leq L_w(s,t)\Psi (t,|u-v|) \] for every \(s,t\in G\), \(u,v\in\mathbb{R}\) and some singularity assumptions. Moreover \(L_w\) is homogeneous with respect to a weight function \(\eta _w\). For the definition of modular spaces, let \(\theta \) be the neutral element of \(G\) and denote by \({\mathcal U}\) a base of measurable neighborhoods of \(\theta \in G\). Let denote by \(L^0(G)\) the space of all the measurable functions \(f:G\to[-\infty,\infty]\), finite a.e. in \(G\). Let \(\rho :L^0(G)\rightarrow [0,+\infty ]\) be a measurable modular functional. From it the following vector subspace of \(L^0(G)\), denoted by \(L^{\rho }(G)\), is defined by \[ L^{\rho }(G)=\{f\in L^0(G):\lim _{\lambda \rightarrow 0^+} \rho (\lambda f)=0\}. \] The subspace \(L^{\rho }(G)\) is the modular space generated by \(\rho \). Two final sections of the paper are devoted respectively to the study of the degree of modular approximation in modular Lipschitz classes and to applications to nets of the so-called nonlinear Mellin convolution operators with weight \(\eta _w(t)=t^{\alpha _w}\), i.e., \[ (M_wf)(s)=\int _0^{\infty }t^{\alpha _w}H_w(ts^{-1})\Gamma (f(t)){dt \over t},\;s\in\mathbb{R}^+. \] The statements of the theorems need too many definitions and notations, so they are not shown here. Many references appear in this nice and complete paper where the general approach used enables the authors to extend previous results by giving a unified treatment of convergence problems for various kinds of nonlinear integral operators in various functional spaces.