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.
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