A knowledge of subsurface temperatures in sedimentary basins, fault zones, volcanic environments and polar ice sheets is of interest for a wide variety of geophysical applications. However, the process of drilling deep boreholes in these environments to provide access for temperature and other measurements invariably disturbs the temperature field around a newly created borehole. Although this disturbance dissipates over time, most temperature measurements are made while the temperature field is still disturbed. Thus, the measurements must be ‘corrected’ for the drilling-disturbance effect if the undisturbed temperature field is to be determined. This paper provides compact analytical solutions for the thermal drilling disturbance based on 1-D (radial) and 2-D (radial and depth) Green's functions (GFs) in cylindrical coordinates. Solutions are developed for three types of boundary conditions (BCs) at the borehole wall: (1) prescribed temperature, (2) prescribed heat flux and (3) a prescribed convective condition. The BC at the borehole wall is allowed to vary both with depth and time. Inclusion of the depth dimension in the 2-D solution allows vertical heat-transfer effects to be quantified in situations where they are potentially important, that is, near the earth's surface, at the bottom of a well and when considering finite-drilling rates. The 2-D solution also includes a radial- and time-dependent BC at the earth's surface to assess the impact of drilling-related infrastructure (drilling pads, mud pits, permanent shelters) on the subsurface temperature field. Latent-heat effects due to the melting and subsequent refreezing of interstitial ice while drilling a borehole through ice-rich permafrost can be included in the GF solution as a moving-plane heat source (or sink) located at the solid–liquid interface. Synthetic examples are provided illustrating the 1-D and 2-D GF solutions. The flexibility of the approach allows the investigation of thermal drilling effects in rock or ice for a wide variety of drilling technologies. Numerical values for the required radial GFs G_R are available through the Advanced Cooperative Arctic Data and Information Service at doi:10.5065/D64F1NS6.