  In this dissertation we study the tractability of the information-based
complexity $n(\varepsilon,d)$ for $d$-variate function approximation problems.
In the deterministic setting for many unweighted problems the curse of
dimensionality holds, that means, for some fixed error tolerance
$\varepsilon>0$ the complexity $n(\varepsilon,d)$ grows exponentially in $d$.
For integration problems one can usually break the curse with the standard
Monte Carlo method. For function approximation problems, however, similar
effects of randomization have been unknown so far.
  The thesis contains results on three more or less stand-alone topics. For an
extended five page abstract, see the section "Introduction and Results".
  Chapter 2 is concerned with lower bounds for the Monte Carlo error for
general linear problems via Bernstein numbers. This technique is applied to the
$L_{\infty}$-approximation of certain classes of $C^{\infty}$-functions, where
it turns out that randomization does not affect the tractability classification
of the problem.
  Chapter 3 studies the $L_{\infty}$-approximation of functions from Hilbert
spaces with methods that may use arbitrary linear functionals as information.
For certain classes of periodic functions from unweighted periodic tensor
product spaces, in particular Korobov spaces, we observe the curse of
dimensionality in the deterministic setting, while with randomized methods we
achieve polynomial tractability.
  Chapter 4 deals with the $L_1$-approximation of monotone functions via
function values. It is known that this problem suffers from the curse in the
deterministic setting. An improved lower bound shows that the problem is still
intractable in the randomized setting. However, Monte Carlo breaks the curse,
in detail, for any fixed error tolerance $\varepsilon>0$ the complexity
$n(\varepsilon,d)$ grows exponentially in $\sqrt{d}$ only.
