  In this paper, we focus on three main objectives related to Hardy-type
inequalities on Cartan-Hadamard manifolds. Firstly, we explore critical
Hardy-type inequalities that contain logarithmic terms, highlighting their
significance. Secondly, we examine the stability of both critical and
subcritical cases of the Hardy inequality. Lastly, we establish two weighted
Hardy-type inequalities where singularities appear at the origin as well as
boundary and we discuss their implications for higher-order operators. Our
results improve upon previous findings and also present new higher-order
versions of these inequalities as additional outcomes.
