The topological complexity of a space \(X,\) denoted as \(\text{TC}(X)\), is defined as the smallest nonnegative integer \(k\) such that \(X\times X\) admits a cover constituted by \(k\) open subsets \(U_1,\dots,U_k\) such that for each \(i\) there is a local homotopy section \(s_i:U_i\rightarrow PX\) of the free path fibration \(\pi :PX\rightarrow X\times X\), \(\alpha \mapsto (\alpha (0),\alpha (1))\) (here \(PX\) denotes the free path space of \(X\) equipped with the compact-open topology). In other words, \(\text{TC}(X)\) is the sectional category (or Schwarz genus) of \(\pi .\) This numerical homotopy invariant was introduced by \textit{M. Farber} [Discrete Comput. Geom. 29 No. 2, 211--221 (2003; Zbl 1038.68130)] in order to study the motion planning problem from a homotopical point of view. An equivariant version of topological complexity for spaces with a given compact group action, called equivariant topological complexity and denoted by \(\text{TC}_G(X),\) was given by \textit{H. Colman} and \textit{M. Grant} [Algebr. Geom. Topol. 12, No. 4, 2299--2316 (2012; Zbl 1260.55007)]. To do this they first established the more general notion of equivariant sectional category of an equivariant map. Then they defined \(\text{TC}_G(X)\) as the equivariant sectional category of the naturally induced equivariant map \(\pi :PX\rightarrow X\times X.\) In the paper under review, the authors establish another equivariant version of topological complexity called invariant topological complexity, denoted by \(\text{TC}^G(X),\) that seems to overcome some difficulties arising in the analysis of impact of symmetries. They first develop the notion of \(A\)-Lusternik-Schnirelmann \(G\)-category of a \(G\)-space \(X\), \({}_A\text{cat}_G(X)\), which is an equivariant version of Clapp-Puppe's invariant \(\mathcal{A}\)-cat(-) [\textit{M. Clapp} and \textit{D. Puppe}, Trans. Am. Math. Soc. 298, 603--620 (1986; Zbl 0618.55003)] when \(\mathcal{A}\) consists just of one space. After giving some interesting properties, including a Whitehead-type approach of \({}_A\text{cat}_G(X)\) and the fact that \(\text{TC}_G(X)={}_{\Delta (X)}\text{cat}_G(X\times X)\) (\(\Delta (X)\) is the diagonal in \(X\times X\)), they define the invariant topological complexity of a \(G\)-space \(X\) as \(\text{TC}(X)^G:={}_{\daleth(X)}\text{cat}_{G\times G}(X\times X),\) where \(\daleth(X)\) is the \(G\times G\) subspace of \(X\times X\) (with respect to the natural induced action) constituted by all the elements of the form \((g\cdot x,g'\cdot x)\) for \(g,g'\in G\) and \(x\in X\). Some interesting properties such as inequalities involving products or the relation with respect to the topological complexity of \(X/G,\) that reflect differences and similarities with respect to the equivariant topological complexity, are given.