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In this work we start by determining all irreducible spherical functions$\Phi$ of any $K $-type associated to the pair $(G,K)=(\SO(4),\SO(3))$. Thefunctions $P=P(u)$ corresponding to the irreducible spherical functions of afixed $K$-type $\pi_\ell$ are appropriately packaged into a sequence of matrixvalued polynomials $(P_w)_{w\ge0}$ of size $(\ell+1)\times(\ell+1)$. Finally weprove that $\widetilde P_w={P_0}^{-1}P_w$ is a sequence of matrix orthogonalpolynomials with respect to a weight matrix $W$. Moreover, we show that $W$admits a second order symmetric hypergeometric operator $\widetilde D$ and afirst order symmetric differential operator $\widetilde E$.  Later, we establish a direct relationship between the spherical functions ofthe $n$-dimensional sphere $S^n\simeq\SO(n+1)/\SO(n)$ and the sphericalfunctions of the $n$-dimensional real projective space$P^n(\mathbb{R})\simeq\SO(n+1)/\mathrm{O}(n)$. Concluding that to find all thespherical functions of one of these pairs is equivalent to do the same it withthe other.  Finally, we study the spherical functions of certain types of the$n$-dimensional sphere $S^{n}\simeq \SO(n+1)/\SO(n)$, for any $n$. Moreprecisely, we give explicitly all the spherical functions whose associatedfunctions $H$ are scalar valued, including those of trivial type, and then westudy the irreducible spherical functions of fundamental type, describing themin terms of matrix valued hypergeometric functions $_2!F_1$. Thereafter, forevery fundamental type we build a sequence of ortogonal matrix valuedpolynomials with respect to a weight $W$, which are associated to the sphericalfunctions. We also prove that, for any $n$, $W$ admits a second order symmetricdifferential operator.

Funciones Esf\'ericas Matriciales Asociadas a las Esferas y a los Espacios Proyectivos Reales