This paper proposes a generalization of the gauge theories that are based on connections with values in a Lie algebra. This objective is attained as follows. Let \(M_ n\) be the base space of the independent variables, and \(R^ N\) the range space of the state variables. It is assumed that an r-parameter Lie group \(G_ r\) acts on \(K=M_ n\times {\mathbb{R}}^ N\) as a group of point transformations. Let \(\{V_ a: a=1,...,r\}\) be a basis of the Lie algebra \(g_ r\) of \(G_ r\) in this representation; thus \(g_ r\) is realized in terms of the maps \(V_ a: \Lambda^ 0(K)\to\Lambda^ 0(K)\). [Here \(\Lambda\) (K) denotes the exterior algebra of differential forms on K.] This situation is denoted by \(g_ r(V_ a;\Lambda^ 0(K))\). It then follows that \(g_ r\) may also be realized by \(g_ r(L_ a;\Lambda (K))\), where \(L_ a\) denotes the Lie derivative with respect to \(V_ a\), and \(\Lambda\) (K) is a domain of \(L_ a\). The operator-valued connection 1-forms are supposed to be represented by \(\Gamma =W^ aL_ a\), where \(W^ a\) are 1-forms on \(G_ r\times K\). The action of \(G_ r\) on \(\Gamma\) gives rise to \('\Gamma ='W^ aL_ a\), where \('W^ a\) is represented in terms of \(W^ a\) by a relation whose structure is reminiscent of that of a standard gauge transformation. These connections define operator-valued curvature 2-forms. A Lie connection is one for which \(L_ 0W^ a=0\), in which case the formalism is very similar to that of classical gauge theories. The resulting theory is applied to action integrals and the associated Euler-Lagrange equations, and to the principle of minimal replacements. The case when \(G_ r\) contains the Poincaré group P on a flat space- time \(M_ 4\) is considered. The Lorentz structure on \(M_ 4\) yields a pseudo-Riemannian structure of signature 2 under the minimal replacement associated with the Lie connection of the local action of P. Field equations for the matter fields and the gauge fields are given for any system of matter fields whose action integral is invariant under the global action of P.