High-quality single crystals of KCo2As2 with the body-centered tetragonal ThCr2Si2 structure were grown using KAs self flux. Structural, magnetic, thermal, and electrical transport properties were investigated. No clear evidence for any phase transitions was found in the temperature range 2-300 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-Grfineisen 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 PdCoO2 and PtCoO2. The in-plane rho(T -> 0) = 0.36 mu Omega cm of KCo2As2 is exceptionally small for this class of compounds. The material also exhibits a 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 KCo2As2 is isotropic and about an order of magnitude smaller than the values for the related compounds SrCo2As2 and BaCo2As2. 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-F. Heat capacity C-p(T) data at low T yield an electronic density of states N(E-F) that is about 36% larger than predicted by the first-principles theory. The C-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). Angle-resolved photoemission spectroscopy measurements are compared with the theoretical predictions of the band structure and Fermi surfaces. Our results show that KCo2As2 provides a new avenue for investigating the physics of high-purity metals.