In this work, we consider the nonlocal obstacle problem with a given obstacle \psi in a bounded Lipschitz domain \Omega in \mathbb{R}^{d} , such that \mathbb{K}_\psi^s=\{v\in H^s_0(\Omega):v\geq\psi \text{ a.e. in }\Omega\}\neq\emptyset , given by u\in\mathbb{K}_\psi^s:\quad\langle\mathcal{L}_au,v-u\rangle\geq\langle F,v-u\rangle\quad\forall v\in\mathbb{K}^s_\psi, for F in H^{-s}(\Omega) , the dual space of the fractional Sobolev space H^s_0(\Omega) , 0<s<1 . The nonlocal operator \mathcal{L}_a:H^s_0(\Omega)\to H^{-s}(\Omega) is defined with a measurable, bounded, strictly positive singular kernel a(x,y):\mathbb{R}^d\times\mathbb{R}^d\to[0,\infty) , by the bilinear form \langle\mathcal{L}_au,v\rangle=\mathrm{P.V.}\int_{\mathbb{R}^d}\int_{\mathbb{R}^d} \tilde{v}(x)(\tilde{u}(x)-\tilde{u}(y))a(x,y) \,dy\,dx=\mathcal{E}_a(u,v), which is a (not necessarily symmetric) Dirichlet form, where \tilde{u},\tilde{v} are the zero extensions of u and v outside \Omega respectively. Furthermore, we show that the fractional operator \tilde{\mathcal{L}}_A=-D^s\cdot AD^s:H^s_0(\Omega)\to H^{-s}(\Omega) defined with the distributional Riesz fractional D^s and with a measurable, bounded matrix A(x) corresponds to a nonlocal integral operator \mathcal{L}_{k_A} with a well-defined integral singular kernel a=k_A . The corresponding s -fractional obstacle problem for \tilde{\mathcal{L}}_A is shown to converge as s\nearrow1 to the obstacle problem in H^1_0(\Omega) with the operator -D\cdot AD given with the classical gradient D . We mainly consider obstacle type problems involving the bilinear form \mathcal{E}_a with one or two obstacles, as well as the N -membranes problem, thereby deriving several results, such as the weak maximum principle, comparison properties, approximation by bounded penalization, and also the Lewy–Stampacchia inequalities. This provides regularity of the solutions, including a global estimate in L^\infty(\Omega) , local Hölder regularity of the solutions when a is symmetric, and local regularity in fractional Sobolev spaces W^{2s,p}_{\mathrm{loc}}(\Omega) and in C^1(\Omega) when \mathcal{L}_a=(-\Delta)^s corresponds to fractional s -Laplacian obstacle type problems u\in\mathbb{K}^s_\psi(\Omega):\quad\int_{\mathbb{R}^d}(D^su-{f})\cdot D^s(v-u)\,dx\geq0\quad\forall v\in\mathbb{K}^s_\psi\text{ for }{f}\in [L^2(\mathbb{R}^d)]^d. These novel results are complemented with the extension of the Lewy–Stampacchia inequalities to the order dual of H^s_0(\Omega) and some remarks on the associated s -capacity and the s -nonlocal obstacle problem for a general \mathcal{L}_a .