We evaluate the performance of several multiterminal detection systems, each of which comprises a central detector and a network of remote sensors. The function of the sensors is to collect data on a random signal source and process this information for transmission to the central detector. Transmission is via noiseless channels of limited capacity, hence data compression is necessary for each sensor. Upon receipt of the transmitted information, the central detector seeks to determine whether the true distribution governing the signal source belongs to a null class II or an alternative class X. System optimization is effected under the classical criterion that stipulates minimization of the type II error rate subject to an upper bound e on the type I error rate. We consider the asymptotic performance 䟭easured by an appropriate error exponents 䟯f five types of systems. The first type has a fixed number of sensors, and processes spatially dependent but temporally independent data of growing sample size in time. Data compression for this type is at rate that tends to zero, and distribution classes II and X each consist of a single element. The second type of system is identical to the first, except for the classes II and X, which are composite The third type of system is a variant of the first which employs fixed-rate data compression. The fourth type is altogether different, in that it employs a variable number of sensors handling independent data of fixed sample size, and intersensor communication is effected by two distinct feedback schemes. The fifth types of system is yet another variant of the first in which data exhibit Markovian dependence in time and are compressed by fixed bit quantizers. In the majority of cases we obtain concise characterizations of the associated error exponents using information-theorectic tools.