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The recently discovered ATi$_3$Bi$_5$ (A=Cs, Rb) exhibit intriguing quantumphenomena including superconductivity, electronic nematicity, and abundanttopological states, which provide promising platforms for studying kagomesuperconductivity, band topology, and charge orders. In this work, wecomprehensively study various properties of ATi$_3$Bi$_5$ includingsuperconductivity under pressure and doping, band topology under pressure,thermal conductivity, heat capacity, electrical resistance, and spin Hallconductivity (SHC) using first-principles calculations. Calculatedsuperconducting transition temperature ($\mathrm{ T_{c}}$) of CsTi$_3$Bi$_5$and RbTi$_3$Bi$_5$ at ambient pressure are about 1.85 and 1.92K. When subjectto pressure, $\mathrm{ T_{c}}$ of CsTi$_3$Bi$_5$ exhibits a special valley anddome shape, which arises from quasi-two-dimensional to three-dimensionalisotropic compression within the context of an overall decreasing trend.Furthermore, $\mathrm{ T_{c}}$ of RbTi$_3$Bi$_5$ can be effectively enhanced upto 3.09K by tuning the kagome van Hove singularities (VHSs) and flat bandthrough doping. Pressure can also induce abundant topological surface states atthe Fermi energy ($\mathrm{E}_{\mathrm{F}}$) and tune VHSs across$\mathrm{E}_{\mathrm{F}}$. Additionally, our transport calculations are inexcellent agreement with recent experiments, confirming the absence of chargedensity wave. Notably, SHC of CsTi$_3$Bi$_5$ can reach as large as 226$\hbar\cdot (e\cdot \Omega \cdot cm) ^{-1} $ at $\mathrm{E}_{\mathrm{F}}$. Ourwork provides a timely and detailed analysis of the rich physical propertiesfor ATi$_3$Bi$_5$, offering valuable insights for further explorations andunderstandings on these intriguing superconducting materials.

Superconducting, topological and transport properties of kagome metals CsTi$ _{3} $Bi$ _{5} $ and RbTi$ _{3} $Bi$ _{5} $