Contrary to MnO2 tantalum capacitors, transient processes in polymer tantalum capacitors after voltage application might result in anomalously high, ampere-level currents when a capacitor might appear as a short circuit for a time that is much greater than a transient time in MnO2 capacitors. In this work, conditions that cause anomalous transients in different types of polymer tantalum capacitors have been analyzed. Related phenomena that included increasing of capacitance and dissipation factors with voltage, parametric surge current test failures without damage to capacitors, and increasing leakage currents at low temperatures are described. It has been shown that anomalous transients increased substantially with applied voltage and with reduction of moisture content in capacitors caused by storage or operation at high temperatures and/or in vacuum. Different types of capacitors exhibited different level of transients and modification of conductive polymers or process of their application might decrease transient leakage currents substantially. For space applications, the risk of failures related to anomalous transients can be mitigated by special testing procedures to select parts with an acceptable level of transients and by voltage derating.