In this paper, a modulus of a curve family is defined by using so-called pseudo-distance functions as follows. Let \(X\) be a connected and open subset of a Riemannian manifold \((M,g)\). Suppose that \(\rho: \Omega(X)\to[0,\infty)\), where \(\Omega(X)\) is the set of all continuous curves in \(X\) parameterized by the unit interval, is invariant under re-parameterization. Then the \(\rho\)-modulus of \({\mathcal A}\subset \Omega(X)\) is defined by \[ {\mathcal M}_\rho({\mathcal A})= \inf_f\|f\|^n_n, \] where \(\|\cdot\|_n^{}\) denotes the \(L^n\)-norm of the Riemannian measure \(d\mu_g\) on \(X\subset M\) induced by \(g\), and the infimum is taken over all non-negative Borel measurable functions \(f\) with the property \[ \int_\gamma f\,ds \geq \rho(\gamma) \] for all \(\gamma \in {\mathcal A}\). This definition leads to an outer measure, which is usually not conformally invariant, but the standard conformally invariant modulus is the special case of the above with \(\rho(\gamma)=1\) for every \(\gamma \in {\mathcal A}\). However, the \(\rho\)-modulus yields a maximal dilatation, which determines how far a mapping is from being conformal. In particular, it is shown that the maximal dilatation defined by using the \(\rho\)-modulus agrees with the maximal dilatation given by the conformally invariant modulus when the distance function \(\rho\) is Riemannian. Further properties of the \(\rho\)-modulus are studied, and several inequalities and identities for the \(\rho\)-modulus are given. As an application, e.g., bounds on volumes of Euclidean balls under quasiconformal mappings are obtained.