We present TEOBResumS, a new effective-one-body (EOB) waveform model for nonprecessing (spin-aligned) and tidally interacting compact binaries. Spin-orbit and spin-spin effects are blended together by making use of the concept of centrifugal EOB radius. The point-mass sector through merger and ringdown is informed by numerical relativity (NR) simulations of binary black holes (BBHs) computed with the SpEC and bam codes. An improved, NR-based phenomenological description of the postmerger waveform is developed. The tidal sector of TEOBResumS describes the dynamics of neutron star binaries up to merger and incorporates a resummed attractive potential motivated by recent advances in the post-Newtonian and gravitational self-force description of relativistic tidal interactions. Equation-of-state-dependent self-spin interactions (monopole-quadrupole effects) are incorporated in the model using leading order post-Newtonian results in a new expression of the centrifugal radius. TEOBResumS is compared to 135 SpEC and 19 bam BBH waveforms. The maximum unfaithfulness to SpEC data ¯F—at design Advanced LIGO sensitivity and evaluated with total mass M with a variance of 10M⊙≤M≤200M⊙—is always below 2.5×10−3 except for a single outlier that grazes the 7.1×10−3 level. When compared to bam data, ¯F is smaller than 0.01 except for a single outlier in one of the corners of the NR-covered parameter space that reaches the 0.052 level. TEOBResumS is also compatible, up to merger, to high-end NR waveforms from binary neutron stars with spin effects and reduced initial eccentricity computed with the bam and thc codes. The data quality of binary neutron star waveforms is assessed via rigorous convergence tests from multiple resolution runs and takes into account systematic effects estimated by using the two independent high-order NR codes. The model is designed to generate accurate templates for the analysis of LIGO-Virgo data through merger and ringdown. We demonstrate its use by analyzing the publicly available data for GW150914.