We investigate to which extent the relevant features of (static) Systemic Risk Measures can be extended to a conditional setting. After providing a general dual representation result, we analyze in greater detail Conditional Shortfall Systemic Risk Measures. In the particular case of such risk measures induced by exponential-type preferences, we provide explicit formulas that also allow us to show a vector-type time consistency property. We also describe systemic risk measures in a robust setting, namely, where no reference probability is assigned and obtain a dual representation for convex robust systemic risk measures adjusted to the financial market. Based on joint papers with M. Burzoni, A. Doldi, F. Zorzi.