Summary: Let \(\mathrm{PG}(\mathbb{F}_q^\nu)\) be the \((\nu-1)\)-dimensional projective space over \(\mathbb{F}_q\) and let \(\Gamma\) be a simple graph of order \(\frac{q^k - 1}{q - 1}\) for some \(k\). A \(2\)-\((\nu,\Gamma, \lambda )\) design over \(\mathbb{F}_q\) is a collection \(\mathcal{B}\) of graphs (blocks) isomorphic to \(\Gamma\) with the following properties: the vertex set of every block is a subspace of \(\mathrm{PG}(\mathbb{F}_q^\nu)\); every two distinct points of \(\mathrm{PG}(\mathbb{F}_q^\nu)\) are adjacent in exactly \(\lambda\) blocks. This new definition covers, in particular, the well-known concept of a \(2\)-\((\nu, k, \lambda)\) design over \(\mathbb{F}_q\) corresponding to the case that \(\Gamma\) is complete. In this study of a foundational nature we illustrate how difference methods allow us to get concrete nontrivial examples of \(\Gamma\)-decompositions over \(\mathbb{F}_2\) or \(\mathbb{F}_3\) for which \(\Gamma\) is a cycle, a path, a prism, a generalized Petersen graph, or a Moebius ladder. In particular, we will discuss in detail the special and very hard case that \(\Gamma\) is complete and \(\lambda = 1\), that is, the Steiner 2-designs over a finite field. Also, we briefly touch the new topic of near resolvable \(2\)-\((\nu, 2, 1)\) designs over \(\mathbb{F}_q\). This study has led us to some (probably new) collateral problems concerning difference sets. Supported by multiple examples, we conjecture the existence of infinite families of \(\Gamma\)-decompositions over a finite field that can be obtained by suitably labeling the vertices of \(\Gamma\) with the elements of a Singer difference set.