A continuous operator $T$ acting on a complex Banach space is a Weyl operator if it is Fredholm of index zero; and $T$ is a Browder operator if it is Fredholm of finite ascent and descent. From these classes we define the Weyl spectrum $\sigma_w(T)$ and the Browder spectrum $\sigma_b(T)$ in the obvious way. We denote by $\pi_{00}(T)$, $\pi_0(T)$ the sets of isolated points $\lambda$ of $\sigma(T)$ such that $0<\dim N(T-\lambda) <\infty$, $\lambda$ is a Riesz point of $T$, respectively. We say that Weyl’s theorem holds for $T$ if $\sigma_w(T) =\sigma(T) \setminus \pi_{00}(T)$; and we say that Browder’s theorem holds for $T$ if $\sigma_w(T) =\sigma(T) \setminus \pi_0(T)$. In most of the results of this paper, $T$ is an operator such that $T$ or $T^*$ has the single-valued extension property (SVEP). In this case, it is shown that the analytic spectral mapping theorem holds for $\sigma_w(T)$, several conditions equivalent to Weyl’s theorem holds for $T$ are given, and it is shown that Browder’s theorem holds for all $f(T)$, $f\in H(\sigma(T))$. These results are applied to study a certain class of operators $\Sigma(X)$. It is shown that, if there exists a function $h\in H(\sigma(T))$ which is identically constant in no connected component of its domain and satisfies $h(T)\in\Sigma(X)$, then Weyl’s theorem holds for $f(T)$ and $f(T^*)$ for all $f\in H(\sigma(T))$. The class $\Sigma(X)$ includes the totally paranormal, the subscalar and some other classes of operators. Thus a unified proof is given for some previously known results.