This paper investigates the performance of high altitude platforms (HAPS) based free-space optical (FSO) communication links including HAPS-to-ground station (downlink), ground-to-HAPS (uplink) and HAPS-to-HAPS (horizontal link) communications. The effects of  attenuation loss, atmospheric turbulence, pointing error and angle-of-arrival (AOA) are taken into account. Also, the application of adaptive optics correction, one of the most effective turbulence mitigation techniques, is analyzed using the Zernike polynomials representation. Closed-form expressions are obtained for probability density function (PDF), cumulative distribution function (CDF), Rytov variance, adaptive optics filter function and outage probability mainly in terms of Meijer’s G function when both no adaptive optics correction is used and adaptive optics correction is applied. Some selected results are presented depending on the various  parameters such as the HAPS altitude, the ratio of vertical and horizontal deviations, beam waist, Zenith angle, height of ground station, receiver aperture diameter, channel state threshold and wind speed. The performance improvement with adaptive optics correction is investigated by removing different Zernike modes. We show that the downlik performance outperforms the uplink, the performance of the horizontal link sharply increases above certain altitude and communication links benefit from the adaptive optics correction up to a certain level in terms of performance improvement.