Many scientific inquiries in natural sciences involve approximating a spherical field -namely a scalar quantity defined over a continuum of directions- from generalised samples of the latter (e.g. directional samples, local averages, etc). Such an approximation task is often carried out by means of a convex optimisation problem, assessing an optimal trade-off between a data-fidelity and regularisation term. To solve this problem numerically, scientists typically discretise the spherical domain by means of quasi-uniform spherical point sets. Finite-difference methods for approximating (pseudo-)differential operators on such discrete domains are however unavailable in general, making it difficult to work with generalised Tikhonov (gTikhonov) or Total Variation (gTV) regularisers, favouring physically admissible spherical fields with smooth and sharp variations respectively. To overcome such limitations, canonical spline-based discretisation schemes have been proposed. In the case of gTikhonov regularisation, the optimality of such schemes has been proven for spherical scattered data interpolation problems with quadratic cost functionals. This result is however too restrictive for most practical purposes, since it is restricted to directional samples and Gaussian noise models. Moreover, a similar optimality result for gTV regularisation is still lacking. In this thesis, we propose a unified theoretical and practical spherical approximation framework for functional inverse problems on the hypersphere. More specifically, we consider recovering spherical fields directly in the continuous domain using penalised convex optimisation problems with gTikhonov or gTV regularisation terms. Our framework is compatible with various measurement types as well as non-differentiable convex cost functionals. Via novel representer theorems, we characterise the solutions of the reconstruction problem for both regularisation strategies. For gTikhonov regularisation, we show that the solution is unique and can be expressed as a linear combination of the sampling linear functionals -modelling the acquisition process- primitived twice with respect to the gTikhonov pseudo-differential operator. For gTV regularisation, we show that the solutions are convex combinations of spherical splines with less innovations than available measurements. We use both results to design canonical spline-based discretisation schemes, exact for gTikhonov regularisation and with vanishing approximation error for gTV regularisation. We propose efficient and provably convergent proximal algorithms to solve the discrete optimisation problems resulting from both discretisation schemes. We illustrate the superiority of our continuous-domain spherical approximation framework over traditional methods on a variety of real and simulated datasets in the fields of meteorology, forestry, radio astronomy and planetary sciences. The sampling functionals, cost functions and regularisation strategies considered in each case are diverse, showing the versatility of both our theoretical framework and algorithmic solutions. In the last part of this thesis finally, we design an efficient and locally convergent algorithm for recovering the spatial innovations of periodic Dirac streams with finite rates of innovation, and propose a recurrent neural-network for boosting spherical approximation methods in the context of real-time acoustic imaging.