Various properties of Toeplitz operators, namely invertibility and Fredholmness, can be studied in terms of an appropriate factorisation of the corresponding symbols. The connections of this factorisation with the solvability of Riemann-Hilbert problems and the corona theorem are discussed, using simple linear algebraic and complex analytic arguments. A class of Toeplitz operators with matrix symbols is presented for which it can be shown that, somewhat surprisingly, semi-Fredholmness is equivalent to invertibility.