  A matroid is a combinatorial structure that captures and generalizes the
algebraic concept of linear independence under a broader and more abstract
framework. Matroids are closely related with many other topics in discrete
mathematics, such as graphs, matrices, codes and projective geometries. In this
work, we define cyclic matroids as matroids over a ground set of size $n$ whose
automorphism group contains an $n$-cycle. We study the properties of such
matroids, with special focus on the minimum size of their basis sets. For this,
we broadly employ two different approaches: the multiple basis exchange
property, and an orbit-stabilizer method, developed by analyzing the action of
the cyclic group of order $n$ on the set of bases. We further present some
applications of our theory to algebra and geometry, presenting connections to
cyclic projective planes, cyclic codes and $k$-normal elements.
