The family of local maximum likelihood (LML) detectors, including the global maximum likelihood (GML) detector, and the family of likelihood ascent search (LAS) detectors are akin to each other and possess common properties significant in both theory and practical multi-input multi-output (MIMO) communications. It is proved that a large MIMO channel possesses the LML characteristic, implying and predicting that a local search detector with likelihood ascent, like a wide-sense sequential LAS (WSLAS) detector, can approach the GML detection. By the replica method, the bit error rate (BER) of an LML detector in the large MIMO channel is obtained. The BER indicates that in the high signal-to-noise ratio (SNR) regime, both the LML and GML detectors achieve the AWGN channel performance when the channel load is as high as up to 1.5086 bits/dimension with an equal-energy distribution, and the channel load can be higher with an unequal-energy distribution. The analytical result is verified by simulation in the equal-energy distribution that the sequential LAS (SLAS) detector, a linear-complexity LML detector, can approach the BER of the NP-hard GML detector. The LML and LAS detectors in the two families are successfully applied to symbol detection in massive antenna MIMO communications and demonstrate the performance near the GML detection. This book chapter reviews the LML and LAS detectors in a unified framework. The focus is on their formulation, relationships, properties, and GML performance in BER and spectral efficiency in large MIMO channels.