Non-smooth dynamical systems emerge in a natural way modeling many real processes and phenomena , for instance , recently piecewise linear differential equations appeared as idealized models of cell activity . Due to that , in these last years , the mathematical community became very interested in understanding the dynamics of these kind of systems . In general , some of the main source of motivation to study non-smooth systems can be found in control theory , impact and friction mechanics nonlinear oscillations , economics , and biology . In this thesis , we are interested in discontinuous piecewise linear differential systems . The study of this particular F(z) amp;#??; sign(x)G(z) $ , where $ z =(x , y) \\in \\mathbb{R}^{?} $ , and $ F $ and $ G $ are linear vector fields in $ \\mathbb{R}^{?} $ or , equivalently , \\begin{eqnarray*} \\begin{array}{*{??}{c}} {} amp; {} amp; {\\dot z = \\left\\{ {\\begin{array}{*{??}{c}} {X(z)\\begin{array}{*{??}{c}} {} amp; {if\\begin{array}{*{??}{c}} {} amp; {x gt; ?,} \\\\ \\end{array}} \\\\ \\end{array}} \\\\ {Y(z)\\begin{array}{*{??}{c}} {} amp; {if\\begin{array}{*{??}{c}} {} amp; {x lt; ?,} \\\\ \\end{array}} \\\\ \\end{array}} \\\\ \\end{array}} \\right.} \\\\ \\end{array}\\ \\end{eqnarray*} where $ X(z) = F(z) amp;#??;G(z) $ and $ Y(z) = F(z)?G(z) $ . The line $\\Sigma = {x = ?} $ is called discontinuity set . Our main goal is to study the maximum number of non-sliding limit cycles that the discontinuous piecewise linear differential system above can have . The systems $ \\dot{z} = X(z) $ and $ \\dot{z}= Y(z) $ are called lateral linear differential systems (or just lateral systems) . From now on in this these , we only consider non-degenerate linear differential systems . System above can be N(R , L) $ . In this thesis , we compute the exact value of $N(L , R) \\leq ? $ always when one of the lateral systems is a saddle of kind $S_{v} , S_{b} , S_{?} $ , a node of kind $N_{r} , N_{b} , N^{*} , iN_{r} , iN_{b} $ , a focus of kind $F_{b} $ and a center $C $ . Particularly , we obtain that $N(L , R) \\leq ? $ in all the above cases . It is easy to see that if one of the lateral linear differential systems is of type $S_{v} , S_{b} , N_{r} , N_{b} , N^{*} , iN_{r} $ , or $iN_{b} $ , then the first return map on the straight line $x = ? $ of system above is not defined . Consequently , system above does not admit non-sliding limit cycles in all these cases . So $N(R , L) = ? $ for the systems having one of these kind of equilibria.\\\\ This thesis deals with the question of the determinacy of the maximum number of limit cycles of some