We define a new model of communication complexity, called the
garden-hose model. Informally, the garden-hose complexity of a
function f:{0,1}^n x {0,1}^n to {0,1} is given by the minimal number
of water pipes that need to be shared between two parties, Alice and
Bob, in order for them to compute the function f as follows: Alice
connects her ends of the pipes in a way that is determined solely by
her input x \in {0,1}^n and, similarly, Bob connects his ends of the
pipes in a way that is determined solely by his input y \in
{0,1}^n. Alice turns on the water tap that she also connected to one
of the pipes. Then, the water comes out on Alice's or Bob's side
depending on the function value f(x,y).  We prove almost-linear lower
bounds on the garden-hose complexity for concrete functions like inner
product, majority, and equality, and we show the existence of
functions with exponential garden-hose complexity. Furthermore, we
show a connection to classical complexity theory by proving that all
functions computable in log-space have polynomial garden-hose
complexity.  We consider a randomized variant of the garden-hose
complexity, where Alice and Bob hold pre-shared randomness, and a
quantum variant, where Alice and Bob hold pre-shared quantum
entanglement, and we show that the randomized garden-hose complexity
is within a polynomial factor of the deterministic garden-hose
complexity. Examples of (partial) functions are given where the
quantum garden-hose complexity is logarithmic in n while the classical
garden-hose complexity can be lower bounded by n^c for constant c>0.
Finally, we show an interesting connection between the garden-hose
model and the (in)security of a certain class of quantum
position-verification schemes.