  High-quality single crystals of KCo$_2$As$_2$ with the body-centered
tetragonal ThCr$_2$Si$_2$ structure were grown using KAs self flux. Structural,
magnetic, thermal, and electrical transport were investigated. No clear
evidence for any phase transitions was found in the temperature range 2 to 300
K. The in-plane electrical resistivity $\rho$ versus temperature $T$ is highly
unusual, showing a $T^4$ behavior below 30 K and an anomalous positive
curvature up to 300 K which is different from the linear behavior expected from
the Bloch-Gr\"uneisen theory for electron scattering by acoustic phonons. This
positive curvature has been previously observed in the in-plane resistivity of
high-conductivity layered delafossites such as PdCoO$_2$ and PtCoO$_2$. The
in-plane $\rho(T\to0) = 0.36~\mu\Omega$ cm of KCo$_2$As$_2$ is exceptionally
small for this class of compounds. The material also exhibits a nearly linear
magnetoresistance at low $T$ which attains a value of about 40% at $T=2$K and
magnetic field $H= 80$ kOe. The magnetic susceptibility $\chi$ of KCo$_2$As$_2$
is isotropic and about an order of magnitude smaller than the values for the
related compounds SrCo$_2$As$_2$ and BaCo$_2$As$_2$. The $\chi$ increases above
100 K which is found from our first-principles calculations to arise from a
sharp peak in the electronic density of states just above the Fermi energy
$E_{\rm F}$. Heat capacity $C_{\rm p}(T)$ data at low $T$ yield an electronic
density of states $N(E_{\rm F})$ that is about 36% larger than predicted by the
first-principles theory. The $C_{\rm p}(T)$ data near room temperature suggest
the presence of excited optic vibration modes which may also be the source of
the positive curvature in $\rho(T)$. Our results show that KCo$_2$As$_2$
provides a new avenue for investigating the physics of high-purity metals.
