This habilitation thesis deals with polynomial system solving through Gröbner bases computations. It focuses on the link between multivariate polynomials and linear recurrence relations satisfied by a multi-indexed sequence for computing Gröbner bases. Our contributions mainly lie on the theoretical and practical aspects on these Gröbner bases computations. First, we present \texttt{msolve}, a new open source \texttt{C} library, for solving polynomial systems using Gröbner bases. Second, we describe new algorithms and complexity estimates for computing Gröbner bases either for a total degree order or the lexicographic one. Then, we present linear algebras-based and polynomial-division-based algorithms for guessing linear recurrences with constant or polynomial coefficients, in generic and structured situations. Finally, we detail our research project for the forthcoming years on these aspects.