Advances in vector sensor technology have created a need for adaptive nonlinear signal processing in the quaternion domain. The main concern of this thesis lies in the issue of analyticity of quaternion-valued nonlinear functions. The Cauchy-Riemann-Fueter (CRF) conditions determine the analyticity in the quaternion domain which proved too strict to be of any practical use. In order to circumvent this problem, split-quaternion nonlinear functions which are analytic componentwise are commonly employed. However, these functions do not fully capture the correlations between dimensions and are not suitable for real-world applications. To address this, the use of fully quaternion nonlinear functions in the derivation of a completely new class of algorithms which takes into consideration the non-commutative aspect of quaternion product is proposed. These fully quaternion functions satisfy the local analyticity condition (LAC) that guarantees the first-order differentiability of the function. This provides a unifying framework for the derivation of gradient based learning algorithms in the quaternion domain which are shown to have the same generic form as their real- and complex-valued counterparts. Unlike existing approaches, this new class of algorithms derived is suitable for the processing of signals with strong component correlations and is further extended to the recurrent neural network (RNN) architecture. Novel algorithms are also derived to improve the computational complexity of quaternion-valued adaptive filters which could be easily extended to incorporate nonlinear functions. A rigorous mathematical analysis provides a basis for the understanding of the convergence and steady-state performance of the proposed algorithms. Simulations over a range of synthetic and real-world signals support the approach taken in the thesis.