Unlike in conventional modulation where information bits are conveyed only through symbols from modulation alphabets defined in the complex plane [e.g., quadrature amplitude modulation (QAM) and phase shift keying (PSK)], in index modulation (IM), additional information bits are conveyed through indexes of certain transmit entities that get involved in the transmission. Transmit antennas in multiantenna systems and subcarriers in multicarrier systems are examples of such transmit entities that can be used to convey additional information bits through indexing. In this paper, we introduce generalized space and frequency IM, where the indexes of active transmit antennas and subcarriers convey information bits. We first introduce IM in the spatial domain, which is referred to as generalized spatial IM (GSIM). For GSIM, where bits are indexed only in the spatial domain, we derive the expression for achievable rate and easy-to-compute upper and lower bounds on this rate. We show that the achievable rate in GSIM can be more than that in spatial multiplexing and analytically establish the condition under which this can happen. It is noted that GSIM achieves this higher rate using fewer transmit radio-frequency (RF) chains compared with spatial multiplexing. We also propose a Gibbs-sampling-based detection algorithm for GSIM and show that GSIM can achieve better bit error rate (BER) performance than spatial multiplexing. For generalized space–frequency IM (GSFIM), where bits are encoded through indexing in both active antennas and subcarriers, we derive the achievable rate expression. Numerical results show that GSFIM can achieve higher rates compared with conventional multiple-input-multiple-output orthogonal frequency division multiplexing (MIMO-OFDM). Moreover, BER results show the potential for GSFIM performing better than MIMO-OFDM.