  The recently discovered kagome superconductor CsV$_3$Sb$_5$ ($T_c \simeq 2.5$
K) has been found to host charge order as well as a non-trivial band topology,
encompassing multiple Dirac points and probable surface states. Such a complex
and phenomenologically rich system is, therefore, an ideal playground for
observing unusual electronic phases. Here, we report on microscopic studies of
its anisotropic superconducting properties by means of transverse-field muon
spin rotation ($\mu$SR) experiments. The temperature dependences of the
in-plane and out-of-plane components of the magnetic penetration depth
$\lambda_{ab}^{-2}(T)$ and $\lambda_{c}^{-2}(T)$ indicate that the
superconducting order parameter exhibits a two-gap ($s+s$)-wave symmetry,
reflecting the multiple Fermi surfaces of CsV3Sb5. The multiband nature of its
superconductivity is further validated by the different temperature dependences
of the anisotropic magnetic penetration depth $\gamma_\lambda(T)$ and upper
critical field $\gamma_{\rm B_{c2}}(T)$, both in close analogy with the well
known two-gap superconductor MgB$_2$. Remarkably, the high value of the
$T_c/\lambda^{-2}(0)$ ratio in both field orientations strongly suggests the
unconventional nature of superconductivity. The relaxation rates obtained from
zero field $\mu$SR experiments do not show noticeable change across the
superconducting transition, indicating that superconductivity does not break
time reversal symmetry.
