The Alon-Jaeger-Tarsi conjecture states that for any finite field F of size at least 4  and any nonsingular  matrix M over F there exists a vector x such that neither x nor Mx has a 0 component. In this talk we discuss the proof of this result for primes larger than 61 and mention further applications of our method about coset covers and additive bases. Joint work with János Nagy and István Tomon.