Harmonic numbers H k = ∑ 0 > j ⩽ k 1 / j ( k = 0 , 1 , 2 , … ) H_{k}=∑ _{0>j⩽ k}1/j\ (k=0,1,2,… ) play important roles in mathematics. In this paper we investigate their arithmetic properties and obtain various basic congruences. Let p > 3 p>3 be a prime. We show that ∑ k = 1 p − 1 H k k 2 k ≡ 0 ( m o d p ) , ∑ k = 1 p − 1 H k 2 ≡ 2 p − 2 ( m o d p 2 ) , ∑ k = 1 p − 1 H k 3 ≡ 6 ( m o d p ) , \begin{equation*} ∑ _{k=1}^{p-1}\frac {H_{k}}{k2^{k}}≡ 0\ (\mathrm {mod} \ p),\ ∑ _{k=1}^{p-1}H_{k}^{2} ≡ 2p-2\ (\mathrm {mod} \ p^{2}), \ ∑ _{k=1}^{p-1}H_{k}^{3}≡ 6\ (\mathrm {mod} \ p),\end{equation*} and ∑ k = 1 p − 1 H k 2 k 2 ≡ 0 ( m o d p ) provided p > 5. \begin{equation*} ∑ _{k=1}^{p-1}\frac {H_{k}^{2}}{k^{2}}≡ 0\ (\mathrm {mod} \ p)\qquad \text {provided }\ p>5. \end{equation*} (In contrast, it is known that ∑ k = 1 ∞ H k / ( k 2 k ) = π 2 / 12 ∑ _{k=1}^{∞ }H_{k}/(k2^{k})=π ^{2}/12 and ∑ k = 1 ∞ H k 2 / k 2 = 17 π 4 / 360 ∑ _{k=1}^{∞ }H_{k}^{2}/k^{2}=17π ^{4}/360 .) Our tools include some sophisticated combinatorial identities and properties of Bernoulli numbers.