The authors prove \(C^{1,\alpha}\)-regularity results for 2D one-phase or two-phase free boundary problems considering Alt-Caffarelli or Alt-Caffarelli-Friedman functionals with variable coefficients. For the one-phase case, they consider the energy functional \(J_{OP}(u,x_{0},r)=\int_{B_{r}(x_{0})}(\sum_{i,j}a_{ij}(x)\frac{\partial u}{\partial x_{i}}\frac{\partial u}{\partial x_{j}}+Q_{OP}(x)\mathbf{1}_{\{u>0\}})dx\) for \(u\in H^{1}(B_{2})\), \(x_{0}\in B_{1}\), \(r\in (0,1)\). Here \(A=(a_{ij})\) is a symmetric matrix of Hölder continuous functions satisfying uniform continuity and coercivity conditions on \(B_{2}\) and \(Q_{OP}\) is a Hölder continuous function which is bounded from above and from below on \(B_{2}\). The authors define the admissible set as \(\mathcal{A}^{+}(B_{r})=\{u\in H^{1}(B_{r}):u\geq 0\) in \(B_{r}\), \(u=0\) on \(B_{r}\setminus B_{r}^{+}\}\), where \(B_{r}^{+}=B_{r}\cap H\), \(H\) being the upper half-plane. They define an almost-minimizer in the upper half-disk \(B_{2}^{+}\) as a function \(u\in\mathcal{A}^{+}(B_{2})\) such that there exist positive constants \(r_{1}\), \(C_{1}\), \(\delta _{1}\) such that for every \(x_{0}\in B_{1}\cap \partial \Omega _{u}\) and \(r\in (0,r_{1})\) \(J_{OP}(u,x_{0},r)\leq (1+C_{1}r^{\delta_{1}})J_{OP}(v,x_{0},r)+C_{1}r^{2+\delta _{1}}\), for every \(v\in \mathcal{A}^{+}(B_{2})\) satisfying \(u=v\)on \(B_{2}\setminus B_{r}(x_{0})\). The main result here proves that if \(u\) is an almost-minimizer of \(J_{OP}\) the free boundary \(B_{1}\cap \partial \Omega _{u}\) is locally the graph of a \(C^{1,\alpha }\)-function. Moreover \(u\) satisfies optimality conditions. For the two-phase case, the authors consider the energy functional \(J_{TP}(u,x_{0},r)=\int_{B_{r}(x_{0})}(\sum_{i,j}a_{ij}(x)\frac{\partial u}{\partial x_{i}}\frac{\partial u}{\partial x_{j}}+Q_{TP}^{+}(x)\mathbf{1}_{\{u>0\}}+Q_{TP}^{-}(x)\mathbf{1}_{\{u<0\}})dx\) for \(u\in H^{1}(B_{2})\), \(x_{0}\in B_{1}\), \(r\in (0,1)\), were \(Q_{TP}^{+}\) and \(Q_{TP}^{-}\) satisfy similar hypotheses as \(Q_{OP}\). They define an almost-minimizer in \(B_{2}\) as a function \(u\in H^{1}(B_{2})\) such that there exist positive constants \( r_{2}\), \(C_{2}\), \(\delta _{2}\) such that for every \(x_{0}\in B_{1}\cap \partial \Omega _{u}\) and \(r\in (0,r_{2})\) \(J_{TP}(u,x_{0},r)\leq (1+C_{2}r^{\delta _{2}})J_{TP}(v,x_{0},r)+C_{2}r^{2+\delta _{2}}\), for every \(v\in H^{1}(B_{2})\) satisfying \(u=v\)on \(B_{2}\setminus B_{r}(x_{0})\). The main result here proves that if \(u\) is a Lipschitz continuous function such that \(u_{\pm }\) are solutions of \(-\operatorname{div}(A\nabla u^{\pm })=f_{\pm }\) in \(\Omega _{u}^{\pm }\cap B_{2}\), where \(f_{\pm }\) are bounded and continuous, and \(u\) is an almost-minimizer of \(J_{TP}\) the free boundaries \(B_{1}\cap \partial \Omega _{u}^{+}\) and \(B_{1}\cap \partial \Omega _{u}^{-}\) are locally the graphs of \(C^{1,\alpha }\)-functions. Moreover \(u\) satisfies optimality conditions on \(\partial \Omega _{u}^{+}\). Here the authors assume that the matrix \(A\) has \(C^{1}\)-regular coefficients. In the two cases, the authors first define Weiss’ boundary adjusted energy functionals for which they prove epiperimetric inequalities. Considering a Lipschitz continuous function \(u\in H^{1}(B_{2})\), a sequence \((x_{n})_{n}\) of points in \( B_{1}\cap \partial \Omega _{u}^{-}\) which converges to some \(x_{0}\in B_{1}\cap \partial \Omega _{u}^{-}\) and a sequence \((r_{n})_{n}\) in \((0,1)\), the authors call \(u_{x_{n},r_{n}}\) a blow-up sequence observe and they that \( u_{x_{n},r_{n}}\) is uniformly-Lipschitz in every compact subset of \(\mathbb{R }^{2}\). They analyze the convergence of blow-up sequences and they prove different properties of the blow-up limits, in the case of an almost-minimizer of the one-phase or two-phase problems. They also define the notion of global solution in both cases. For the proof of the regularity result in both cases, the authors prove, for every \(x_{0}\in B_{1}\cap \partial \Omega _{u}^{-}\), the existence of a unique blow-up limit \(u_{x_{n}} \) of the almost-minimizer \(u\) at \(x_{0}\) and they prove some flatness estimate for the free-boundary.